% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of General Investigations of Curved Surfaces%
% of 1827 and 1825, by Karl Friedrich Gauss                               %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.net                          %
%                                                                         %
%                                                                         %
% Title: General Investigations of Curved Surfaces of 1827 and 1825       %
%                                                                         %
% Author: Karl Friedrich Gauss                                            %
%                                                                         %
% Translator: James Caddall Morehead                                      %
%             Adam Miller Hiltebeitel                                     %
%                                                                         %
% Release Date: July 25, 2011 [EBook #36856]                              %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
%                                                                         %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

\def\ebook{36856}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                  %%
%% Packages and substitutions:                                      %%
%%                                                                  %%
%% book:     Required.                                              %%
%% inputenc: Latin-1 text encoding. Required.                       %%
%% fontenc:  Font encoding, to hyphenate accented words. Required.  %%
%%                                                                  %%
%% ifthen:   Logical conditionals. Required.                        %%
%%                                                                  %%
%% amsmath:  AMS mathematics enhancements. Required.                %%
%% amssymb:  Additional mathematical symbols. Required.             %%
%%                                                                  %%
%% alltt:    Fixed-width font environment. Required.                %%
%% array:    Enhanced tabular features. Required.                   %%
%%                                                                  %%
%% indentfirst: Indent first line of section. Required.             %%
%% footmisc: Start footnote numbering on each page. Required.       %%
%%                                                                  %%
%% graphicx: Standard interface for graphics inclusion. Required.   %%
%%                                                                  %%
%% calc:     Length calculations. Required.                         %%
%%                                                                  %%
%% fancyhdr: Enhanced running headers and footers. Required.        %%
%%                                                                  %%
%% geometry: Enhanced page layout package. Required.                %%
%% hyperref: Hypertext embellishments for pdf output. Required.     %%
%%                                                                  %%
%%                                                                  %%
%% Producer's Comments:                                             %%
%%                                                                  %%
%%   OCR text for this ebook was obtained on June 24, 2011, from    %%
%%   http://www.archive.org/details/generalinvestigaOOgaus.         %%
%%                                                                  %%
%%   Minor changes to the original are noted in this file in three  %%
%%   ways:                                                          %%
%%     1. \Typo{}{} for typographical corrections, showing original %%
%%        and replacement text side-by-side.                        %%
%%     2. \Add{} for inconsistent/missing punctuation.              %%
%%     3. [** TN: Note]s for lengthier or stylistic comments.       %%
%%                                                                  %%
%%                                                                  %%
%% Compilation Flags:                                               %%
%%                                                                  %%
%%   The following behavior may be controlled by boolean flags.     %%
%%                                                                  %%
%%   ForPrinting (true by default):                                 %%
%%   If false, compile a screen optimized file (one-sided layout,   %%
%%   blue hyperlinks). Default: print-optimized PDF file: Larger    %%
%%   text block, two-sided layout, black hyperlinks.                %%
%%                                                                  %%
%%   Modernize (true by default):                                   %%
%%   Set book and journal article titles in italics; normalize      %%
%%   miscellaneous abbreviations and displayed equation punctuation;%%
%%   use square root signs instead of surds.                        %%
%%                                                                  %%
%%                                                                  %%
%% PDF pages: 153 (if ForPrinting set to true)                      %%
%% PDF page size: US Letter (8.5 x 11")                             %%
%% PDF bookmarks: created, point to ToC entries                     %%
%% PDF document info: filled in                                     %%
%% Images: 5 pdf diagrams                                           %%
%%                                                                  %%
%% Summary of log file:                                             %%
%% * Several slightly over-full hboxes; no visual issues.           %%
%%                                                                  %%
%% Compile History:                                                 %%
%%                                                                  %%
%% July, 2011: (Andrew D. Hwang)                                    %%
%%             texlive2007, GNU/Linux                               %%
%%                                                                  %%
%% Command block:                                                   %%
%%                                                                  %%
%%     pdflatex x2                                                  %%
%%                                                                  %%
%%                                                                  %%
%% July 2011: pglatex.                                              %%
%%   Compile this project with:                                     %%
%%   pdflatex 36856-t.tex ..... TWO times                           %%
%%                                                                  %%
%%   pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6)                 %%
%%                                                                  %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\listfiles
\documentclass[12pt,leqno]{book}[2005/09/16]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[latin1]{inputenc}[2006/05/05]
\usepackage[T1]{fontenc}[2005/09/27]

\usepackage{ifthen}[2001/05/26]  %% Logical conditionals

\usepackage{amsmath}[2000/07/18] %% Displayed equations
\usepackage{amssymb}[2002/01/22] %% and additional symbols

\usepackage{alltt}[1997/06/16]   %% boilerplate, credits, license
\usepackage{array}[2005/08/23]   %% extended array/tabular features

\usepackage{indentfirst}[1995/11/23]
\usepackage[perpage,symbol]{footmisc}[2005/03/17]

\usepackage{graphicx}[1999/02/16]%% For diagrams

\usepackage{calc}[2005/08/06]

\usepackage{fancyhdr} %% For running heads

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Interlude:  Set up PRINTING (default) or SCREEN VIEWING %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% ForPrinting=true                     false (default)
% Asymmetric margins                   Symmetric margins
% 1 : 1.62 text block aspect ratio     3 : 4 text block aspect ratio
% Black hyperlinks                     Blue hyperlinks
% Start major marker pages recto       No blank verso pages
%
\newboolean{ForPrinting}

%% UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION of the text %%
\setboolean{ForPrinting}{true}

% Modernize notation: Set paper titles in italics; normalize punctuation
% and other minor formatting; use square root signs instead of surds.
\newboolean{Modernize}
%% COMMENT the next line to reproduce the original book. (Typographical
%% Corrections will be effected either way.)
\setboolean{Modernize}{true}

%% Initialize values to ForPrinting=false
\newcommand{\Margins}{hmarginratio=1:1}     % Symmetric margins
\newcommand{\HLinkColor}{blue}              % Hyperlink color
\newcommand{\PDFPageLayout}{SinglePage}
\newcommand{\TransNote}{Transcriber's Note}
\newcommand{\TransNoteCommon}{%
  The camera-quality files for this public-domain ebook may be
  downloaded \textit{gratis} at
  \begin{center}
    \texttt{www.gutenberg.org/ebooks/\ebook}.
  \end{center}

  This ebook was produced using scanned images and OCR text generously
  provided by the Boston University Library through the Internet
  Archive.
  \bigskip

  Minor typographical corrections and presentational changes have been
  made without comment. Punctuation has been regularized, but may be
  easily reverted to match the original; changes are documented in the
  \LaTeX\ source file.
  \bigskip

  Displayed equations referenced in the translators' notes
  (pp.~\pageref{notes:1827}~\textit{ff.}\ and
  pp.~\pageref{notes:1825}~\textit{ff.})\ are marked with~\NoteMark.
  In-line page number references may be off by one.
  \bigskip
}

\newcommand{\TransNoteText}{%
  \TransNoteCommon

  This PDF file is optimized for screen viewing, but may be recompiled
  for printing. Please consult the preamble of the \LaTeX\ source file
  for instructions and other particulars.
}
%% Re-set if ForPrinting=true
\ifthenelse{\boolean{ForPrinting}}{%
  \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins
  \renewcommand{\HLinkColor}{black}         % Hyperlink color
  \renewcommand{\PDFPageLayout}{TwoPageRight}
  \renewcommand{\TransNote}{Transcriber's Note}
  \renewcommand{\TransNoteText}{%
    \TransNoteCommon

    This PDF file is optimized for printing, but may be recompiled for
    screen viewing. Please consult the preamble of the \LaTeX\ source
    file for instructions and other particulars.
  }
}{% If ForPrinting=false, don't skip to recto
  \renewcommand{\cleardoublepage}{\clearpage}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%  End of PRINTING/SCREEN VIEWING code; back to packages  %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\ifthenelse{\boolean{ForPrinting}}{%
  \setlength{\paperwidth}{8.5in}%
  \setlength{\paperheight}{11in}%
% ~1:1.67
  \usepackage[body={5.75in,9in},\Margins]{geometry}[2002/07/08]
}{%
  \setlength{\paperwidth}{6in}%
  \setlength{\paperheight}{8in}%
  \raggedbottom
% ~3:4
  \usepackage[body={5.75in,7in},\Margins,includeheadfoot]{geometry}[2002/07/08]
}

\providecommand{\ebook}{00000}    % Overridden during white-washing
\usepackage[pdftex,
  hyperref,
  hyperfootnotes=false,
  pdftitle={The Project Gutenberg eBook \#\ebook: General Investigations of Curved Surfaces},
  pdfauthor={Karl Friedrich Gauss},
  pdfkeywords={Boston University, The Internet Archive, James Caddall Morehead, Adam Miller Hiltbeitel, Andrew D. Hwang},
  pdfstartview=Fit,    % default value
  pdfstartpage=1,      % default value
  pdfpagemode=UseNone, % default value
  bookmarks=true,      % default value
  linktocpage=false,   % default value
  pdfpagelayout=\PDFPageLayout,
  pdfdisplaydoctitle,
  pdfpagelabels=true,
  bookmarksopen=true,
  bookmarksopenlevel=0,
  colorlinks=true,
  linkcolor=\HLinkColor]{hyperref}[2007/02/07]


%% Fixed-width environment to format PG boilerplate %%
\newenvironment{PGtext}{%
\begin{alltt}%****
%\fontsize{8.1}{9}\ttfamily\selectfont}%
\fontsize{9.2}{11}\ttfamily\selectfont}%
{\end{alltt}}

%% Miscellaneous global parameters %%
% No hrule in page header
\renewcommand{\headrulewidth}{0pt}

% For extra-loose spacing in bibliography entries
\newcommand{\Loosen}{\spaceskip0.375em plus 0.25em minus 0.125em}
\hyphenation{géo-dé-siques vor-ge-schriebenen vor-ge-schriebenem}

% Globally loosen the spacing
\setlength{\emergencystretch}{1em}

% Scratch pad for length calculations
\newlength{\TmpLen}

%% Running heads %%
\newcommand{\FlushRunningHeads}{\clearpage\fancyhf{}\cleardoublepage}
\newcommand{\InitRunningHeads}{%
  \setlength{\headheight}{15pt}
  \pagestyle{fancy}
  \thispagestyle{empty}
  \ifthenelse{\boolean{ForPrinting}}
             {\fancyhead[RO,LE]{\thepage}}
             {\fancyhead[R]{\thepage}}
}

\newcommand{\SetRunningHeads}[2]{%
  \fancyhead[CE]{\textsc{\MakeLowercase{#1}}}
  \fancyhead[CO]{\textsc{\MakeLowercase{#2}}}
}

\newcommand{\BookMark}[2]{\phantomsection\pdfbookmark[#1]{#2}{#2}}

%% Major document divisions %%
\newcommand{\PGBoilerPlate}{%
  \pagenumbering{Alph}
  \pagestyle{empty}
  \BookMark{0}{PG Boilerplate.}
}
\newcommand{\FrontMatter}{%
  \cleardoublepage
  \frontmatter
  \BookMark{-1}{Front Matter.}
}
\newcommand{\MainMatter}{%
  \FlushRunningHeads
  \InitRunningHeads
  \mainmatter
  \BookMark{-1}{Main Matter.}
}
\newcommand{\BackMatter}{%
  \FlushRunningHeads
  \InitRunningHeads
  \backmatter
  \BookMark{-1}{Back Matter.}
}
\newcommand{\PGLicense}{%
  \FlushRunningHeads
  \pagenumbering{Roman}
  \InitRunningHeads
  \BookMark{-1}{PG License.}
  \SetRunningHeads{License.}{License.}
}

%% Sectional units %%
\newcommand{\SectTitle}[1]{%
  \section*{\centering\normalfont\large\MakeUppercase{#1}}
}

% Keep track internally of which paper we're typesetting
\newcommand{\PaperNo}{}
% \Paper{1827/1825}
% Clear running heads and set up anchors for each paper's title page
\newcommand{\Paper}[1]{%
  \FlushRunningHeads
  \BookMark{0}{Paper of #1}%
  \renewcommand{\PaperNo}{#1}
  \phantomsection\label{paper:#1}%
}

% Print heading at the start of each paper
\newcommand{\PaperTitle}[1]{%
  \cleardoublepage
  \InitRunningHeads
  % ** Project-dependent logic
  \ifthenelse{\equal{\PaperNo}{1827}}{%
    \SetRunningHeads
    {Karl Friedrich Gauss}
    {General Investigations of Curved Surfaces}
  }{% Else \PaperNo = 1825
    \SetRunningHeads
    {Karl Friedrich Gauss}
    {New General Investigations of Curved Surfaces [1825]}
  }
  \thispagestyle{empty}
  \begin{center}
    #1\\
    \tb
  \end{center}
}

\newcommand{\Introduction}{%
  \FlushRunningHeads
  \InitRunningHeads
  \BookMark{0}{Introduction.}
  \SetRunningHeads{Introduction.}{Introduction.}
  \SectTitle{INTRODUCTION}
}

%% ToC formatting %%
\newcommand{\Contents}{%
  \FlushRunningHeads
  \BookMark{0}{Contents.}
  \SectTitle{CONTENTS}
  \vfil
  \noindent\hfill{\footnotesize PAGE}
}

%\ToCSect{Title}{xref}
\newcommand{\ToCLine}[2]{%
  \settowidth{\TmpLen}{999}%
  \noindent\strut\parbox[b]{\textwidth-\TmpLen}{%
    \hangindent2em #1\dotfill}%
  \makebox[\TmpLen][r]{\pageref{#2}}%

  \vfil
}

% Numbered units
\newcommand{\Article}[1]{%
  \section*{\centering\normalsize #1}
  \phantomsection\label{art:\PaperNo.#1}
}

% and links
\newcommand{\Art}[2][\PaperNo]{\hyperref[art:#1.#2.]{Art.~#2}}
\newcommand{\Arts}[3][,~]{%
  Arts.~\hyperref[art:\PaperNo.#2.]{#2}#1\hyperref[art:\PaperNo.#3.]{#3}%
}

\newcommand{\Abstract}[1]{%
  \FlushRunningHeads
  \InitRunningHeads
  \SetRunningHeads{Gauss's Abstract.}{Gauss's Abstract.}
  \BookMark{0}{Gauss's Abstract.}
  \phantomsection\label{abstract}
  \begin{center}
    #1\\
    \tb
  \end{center}
}

% Translators' notes
\newcommand{\Notes}[1]{%
  \clearpage
  \InitRunningHeads
  \SetRunningHeads{Notes.}{Notes.}
  \BookMark{0}{Notes (\PaperNo)}
  \label{notes:\PaperNo}
  \section*{\centering\normalsize\normalfont NOTES.\Caveat}
}

% Transcriber's caution
\newcommand{\Caveat}{%
  \footnote[3]{Line number references in the translators' notes
    are omitted. Descriptions such as ``top of page~$n$'' are
    retained, but may not match this ebook's pagination. [\textit{Transcriber}]}}

\newcommand{\Par}[1]{\paragraph*{\normalfont\indent #1}}

% Invisible manual anchor for \LineRef
\newcommand{\Note}[1]{\phantomsection\label{note:#1}}
% Mark equations referred to by notes; [2] -> \dag
\newcommand{\NoteMark}{\footnotemark[2]}

% Macro discards its third argument, including the original line number
%\LineRef[xref]{Article Number}{Original text}
\newcommand{\LineRef}[3][]{%
  \ifthenelse{\equal{#1}{}}{%
    \hyperref[art:\PaperNo.#2.]{Art.~#2, p.~\pageref{art:\PaperNo.#2.}}%
  }{%
    \hyperref[note:#1]{Art.~#2, p.~\pageref{note:#1}}%
  }%
}

%% One- and two-off macros %%
% Hard-coded hyperref, discards argument
\newcommand{\LineRefs}[1]{Art.~2, pp.~\pageref{note:4}--\pageref{note:5}}

% End of Introduction
\newcommand{\Signature}[4]{%
  \medskip%
  \null\hfill\textsc{#1}\hspace*{\parindent} \\
  \settowidth{\TmpLen}{\scshape #3}
  \begin{minipage}{\TmpLen}
    \centering\scshape #2 \\ #3 \\ #4\par
  \end{minipage}
  \newpage
}

% Indented, three-line formatting of Gauss's paper titles in Introduction
\newcommand{\Publication}[3]{%
  #1 \\
  \hspace*{3\parindent}#2 \\
  \hspace*{4\parindent}#3\ignorespaces
}

% For table of towns and angular separations
\newcommand{\Dotrow}[2]{\makebox[\TmpLen][s]{#1\dotfill $#2$}}

% Italicized text
\newenvironment{Theorem}[1][Theorem. ]{%
  \textsc{#1}\itshape\ignorespaces}{\upshape\par\medskip}

%% Diagrams %%
\newcommand{\Graphic}[2]{%
  \phantomsection\label{fig:#2}%
  \includegraphics[width=#1]{./images/#2.pdf}%
}
% \Figure{file}{width}
\newcommand{\Figure}[2][0.8\textwidth]{%
  \begin{figure}[hbt!]
    \centering
    \Graphic{#1}{#2}
  \end{figure}\ignorespaces%
}

\newcommand{\BibliographyPage}{%
  \FlushRunningHeads
  \null\vfill
  \phantomsection\label{biblio}
  \BookMark{0}{Bibliography.}
  {\noindent\LARGE BIBLIOGRAPHY}
  \vfill
  \newpage
}

%[** TN: Original bibliography is set in two columns; discarding formatting]
\newenvironment{Bibliography}[1]{%
  \FlushRunningHeads
  \InitRunningHeads
  \SetRunningHeads{Bibliography.}{Bibliography.}
  \section*{\centering\normalsize\normalfont BIBLIOGRAPHY.}

  #1

  \begin{description}\setlength{\parindent}{1.5em}%
}{%
  \end{description}
}

\newcommand{\Author}[1]{\item{\normalfont #1}}

% Decorative rule
\newcommand{\tb}[1][0.75in]{\rule{#1}{0.5pt}}

%% Corrections/modernizations. %%
% Errors found during digitization
\newcommand{\Typo}[2]{#2}
% For list of Corrigenda et addenda in the original
\newcommand{\Erratum}[2]{#2}

\ifthenelse{\boolean{Modernize}}{%
  \newcommand{\Chg}[2]{#2}
  \newcommand{\Add}[1]{#1}
  % Use "page~xx" for all page refs (original sometimes uses "p.~xx")
  \newcommand{\Pgref}[2][page]{\hyperref[#2]{page~\pageref*{#2}}}
  \newcommand{\Pageref}[2][page]{\Pgref{page:#2}}
  % Paper titles not italicized in the original
  \newcommand{\Title}[1]{\Loosen\textit{#1}}
  \newcommand{\Sqrt}[2][]{\:\sqrt[#1]{#2}}
  \newcommand{\SQRT}[2][]{\:\sqrt[#1]{#2}}
}{% Modernize = false
  \newcommand{\Chg}[2]{#1}
  \newcommand{\Add}[1]{}
  \newcommand{\Pgref}[2][page]{\hyperref[#2]{#1~\pageref*{#2}}}
  \newcommand{\Pageref}[2][page]{\Pgref[#1]{page:#2}}
  \newcommand{\Title}[1]{\Loosen#1}
  % Use surd sign...
  \let\oldsqrt=\sqrt%
  \renewcommand{\sqrt}[2][]{\:\oldsqrt[#1]{\vphantom{#2}}\!#2}
  % ... with parentheses or curly braces around radicand
  \newcommand{\Sqrt}[2][]{\sqrt[#1]{\left(#2\right)}}
  \newcommand{\SQRT}[2][]{\sqrt[#1]{\biggl\{#2\biggr\}}}
}

% Page separators
\newcommand{\PageSep}[1]{\phantomsection\label{page:#1}\ignorespaces}

% Miscellaneous conveniences
\newcommand{\eg}{\textit{e.\,g.}}
\newcommand{\ie}{\textit{i.\,e.}}
\newcommand{\Cf}{\textit{Cf}}
\newcommand{\QEA}{\textit{Q.\;E.\;A.}}

%% Miscellaneous mathematical formatting %%
\DeclareInputMath{176}{{}^\circ}
\DeclareInputMath{183}{\cdot}

\DeclareMathOperator{\cosec}{cosec}
\DeclareMathOperator{\Area}{Area}

\newcommand{\dd}{\partial}
\renewcommand{\L}{\mathfrak{L}}
\newcommand{\Z}{\phantom{0}}
\newcommand{\Neg}{\phantom{-}}
\newcommand{\Tag}[1]{%
  \tag*{\ensuremath{#1}}
}

%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%% PG BOILERPLATE %%
\PGBoilerPlate
\begin{center}
\begin{minipage}{\textwidth}
\small
\begin{PGtext}
The Project Gutenberg EBook of General Investigations of Curved Surfaces
of 1827 and 1825, by Karl Friedrich Gauss

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net


Title: General Investigations of Curved Surfaces of 1827 and 1825

Author: Karl Friedrich Gauss

Translator: James Caddall Morehead
            Adam Miller Hiltebeitel

Release Date: July 25, 2011 [EBook #36856]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
\end{PGtext}
\end{minipage}
\end{center}
\newpage
%% Credits and transcriber's note %%
\begin{center}
\begin{minipage}{\textwidth}
\begin{PGtext}
Produced by Andrew D. Hwang, with special thanks to Brenda Lewis.
\end{PGtext}
\end{minipage}
\vfill
\end{center}

\begin{minipage}{0.85\textwidth}
\small
\BookMark{0}{Transcriber's Note.}
\subsection*{\centering\normalfont\scshape%
\normalsize\MakeLowercase{\TransNote}}%

\raggedright
\TransNoteText
\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
\PageSep{i}
\FrontMatter
\begin{center}
{\LARGE\scshape Karl Friedrich Gauss} \\
\tb
\bigskip

\LARGE\scshape General Investigations \\
{\footnotesize OF} \\
Curved Surfaces \\
{\footnotesize OF} \\
\large 1827 and 1825
\vfill

\normalsize
TRANSLATED WITH NOTES \\
{\scriptsize AND A} \\[4pt]
BIBLIOGRAPHY \\[4pt]
{\scriptsize BY \\[4pt]
JAMES CADDALL MOREHEAD, A.M., M.S., and ADAM MILLER HILTEBEITEL, A.M. \\[4pt]
J. S. K. FELLOWS IN MATHEMATICS IN PRINCETON UNIVERSITY}
\vfill\vfill

THE PRINCETON UNIVERSITY LIBRARY \\
1902
\end{center}
\newpage
\PageSep{ii}
\null\vfill
\begin{center}
\scriptsize
Copyright, 1902, by \\
\textsc{The Princeton University Library}
\vfill
\textit{C. S. Robinson \&~Co., University Press \\
Princeton, N. J.}
\end{center}
\PageSep{iii}


\Introduction

In 1827 Gauss presented to the Royal Society of Göttingen his important paper on
the theory of surfaces, which seventy-three years afterward the eminent French
geometer, who has done more than any one else to propagate these principles, characterizes
as one of Gauss's chief titles to fame, and as still the most finished and useful
introduction to the study of infinitesimal geometry.\footnote
  {G. Darboux, Bulletin des Sciences Math. Ser.~2, vol.~24, page~278, 1900.}
This memoir may be called:
\Title{General Investigations of Curved Surfaces}, or the Paper of~1827, to distinguish it
from the original draft written out in~1825, but not published until~1900. A list of
the editions and translations of the Paper of~1827 follows. There are three editions
in Latin, two translations into French, and two into German. The paper was originally
published in Latin under the title:

I\textit{a}. \Publication{Disquisitiones generales circa superficies curvas}
{auctore Carolo Friderico Gauss\Add{.}}
{Societati regiæ oblatæ D.~8.~Octob.~1827}, \\
and was printed in: Commentationes societatis regiæ scientiarum Gottingensis recentiores,
Commentationes classis mathematicæ. Tom.~VI. (ad~a.\ 1823--1827). Gottingæ,
1828, pages~99--146. This sixth volume is rare; so much so, indeed, that the British
Museum Catalogue indicates that it is missing in that collection. With the signatures
changed, and the paging changed to pages~1--50, I\textit{a}~also appears with the title page
added:

I\textit{b}. \Publication{Disquisitiones generales circa superficies curvas}
{auctore Carolo Friderico Gauss.}
{Gottingæ. Typis Dieterichianis. 1828.}

II\@. In Monge's \Title{Application de l'analyse à la géométrie}, fifth edition, edited by
Liouville, Paris, 1850, on pages 505--546, is a reprint, added by the Editor, in Latin
under the title: \Title{Recherches sur la théorie générale des surfaces courbes}; Par M.
C.-F. Gauss.
\PageSep{iv}

III\textit{a}. A third Latin edition of this paper stands in: Gauss, \Title{Werke, Herausgegeben
von der Königlichen Gesellschaft der Wissenschaften zu Göttingen}, Vol.~4, Göttingen,
1873, pages~217--258, without change of the title of the original paper~(I\textit{a}).

III\textit{b}. The same, without change, in Vol.~4 of Gauss, \Title{Werke}, Zweiter Abdruck,
Göttingen,~1880.

IV\@. A French translation was made from Liouville's edition,~II, by Captain
Tiburce Abadie, ancien élève de l'École Polytechnique, and appears in Nouvelles
Annales de Mathématique, Vol.~11, Paris,~1852, pages~195--252, under the title:
\Title{Recherches générales sur les surfaces courbes}; Par M.~Gauss. This latter also
appears under its own title.

V\textit{a}. Another French translation is: \Title{Recherches Générales sur les Surfaces
Courbes}. Par M. C.-F. Gauss, traduites en français, suivies de notes et d'études
sur divers points de la Théorie des Surfaces et sur certaines classes de Courbes, par
M. E. Roger, Paris, 1855.

V\textit{b}. The same. Deuxième Edition. Grenoble (or Paris), 1870 (or 1871), 160~pages.

VI\@. A German translation is the first portion of the second part, namely, pages
198--232, of: Otto Böklen, \Title{Analytische Geometrie des Raumes}, Zweite Auflage, Stuttgart,
1884, under the title (on page~198): \Title{Untersuchungen über die allgemeine Theorie
der krummen Flächen}. Von C.~F. Gauss. On the title page of the book the second
part stands as: \Title{Disquisitiones generales circa superficies curvas} von C.~F. Gauss, ins
Deutsche übertragen mit Anwendungen und Zusätzen\dots.

VII\textit{a}. A second German translation is No.~5 of Ostwald's Klassiker der exacten
Wissenschaften: \Title{Allgemeine Flächentheorie} (\Title{Disquisitiones generales circa superficies
curvas}) von Carl Friedrich Gauss, (1827). Deutsch herausgegeben von A.~Wangerin.
Leipzig, 1889. 62~pages.

VII\textit{b}. The same. Zweite revidirte Auflage. Leipzig, 1900. 64~pages.

The English translation of the Paper of~1827 here given is from a copy of the
original paper,~I\textit{a}; but in the preparation of the translation and the notes all the
other editions, except~V\textit{a}, were at hand, and were used. The excellent edition of
Professor Wangerin,~VII, has been used throughout most freely for the text and
notes, even when special notice of this is not made. It has been the endeavor of
the translators to retain as far as possible the notation, the form and punctuation of
the formulæ, and the general style of the original papers. Some changes have been
made in order to conform to more recent notations, and the most important of those
are mentioned in the notes.
\PageSep{v}

%[** TN: Paragraph not indented in the original]
The second paper, the translation of which is here given, is the abstract (Anzeige)
which Gauss presented in German to the Royal Society of Göttingen, and which was
published in the Göttingische gelehrte Anzeigen. Stück~177. Pages 1761--1768. 1827.
November~5. It has been translated into English from pages 341--347 of the fourth
volume of Gauss's Works. This abstract is in the nature of a note on the Paper of~1827,
and is printed before the notes on that paper.

Recently the eighth volume of Gauss's Works has appeared. This contains on
pages 408--442 the paper which Gauss wrote out, but did not publish, in~1825. This
paper may be called the \Title{New General Investigations of Curved Surfaces}, or the Paper
of~1825, to distinguish it from the Paper of~1827. The Paper of~1825 shows the
manner in which many of the ideas were evolved, and while incomplete and in some
cases inconsistent, nevertheless, when taken in connection with the Paper of~1827,
shows the development of these ideas in the mind of Gauss. In both papers are
found the method of the spherical representation, and, as types, the three important
theorems: The measure of curvature is equal to the product of the reciprocals of the
principal radii of curvature of the surface, The measure of curvature remains unchanged
by a mere bending of the surface, The excess of the sum of the angles of a geodesic
triangle is measured by the area of the corresponding triangle on the auxiliary sphere.
But in the Paper of~1825 the first six sections, more than one-fifth of the whole paper,
take up the consideration of theorems on curvature in a plane, as an introduction,
before the ideas are used in space; whereas the Paper of~1827 takes up these ideas
for space only. Moreover, while Gauss introduces the geodesic polar coordinates in
the Paper of~1825, in the Paper of~1827 he uses the general coordinates, $p$,~$q$, thus
introducing a new method, as well as employing the principles used by Monge and
others.

The publication of this translation has been made possible by the liberality of
the Princeton Library Publishing Association and of the Alumni of the University
who founded the Mathematical Seminary.

\Signature{H. D. Thompson.}
{Mathematical Seminary,}
{Princeton University Library,}
{January 29, 1902.}
\PageSep{vi}
%[Blank page]
\PageSep{vii}


\Contents

\ToCLine{Gauss's Paper of 1827, General Investigations of Curved Surfaces}
{paper:1827} % 1

\ToCLine{Gauss's Abstract of the Paper of 1827}
{abstract} % 45

\ToCLine{Notes on the Paper of 1827}
{notes:1827} % 51

\ToCLine{Gauss's Paper of 1825, New General Investigations of Curved Surfaces}
{paper:1825} % 79

\ToCLine{Notes on the Paper of 1825}
{notes:1825} % 111

\ToCLine{Bibliography of the General Theory of Surfaces}
{biblio} % 115
\PageSep{viii}
%[Blank page]
\PageSep{1}
\MainMatter
\Paper{1827}
%\thispagestyle{empty}
\begin{center}
\LARGE
DISQUISITIONES GENERALES
\vfil
{\normalsize CIRCA}
\vfil
\LARGE
SUPERFICIES CURVAS
\vfil
{\normalsize AUCTORE} \\[8pt]
CAROLO FRIDERICO GAUSS
\vfil

\footnotesize
SOCIETATI REGIAE OBLATAE D.~8.~OCTOB.~1827
\vfil

\tb \\
\medskip

COMMENTATIONES SOCIETATIS REGIAE SCIENTIARUM \\[4pt]
GOTTINGENSIS RECENTIORES\@. VOL.~VI\@. GOTTINGAE MDCCCXXVIII \\
\tb
\vfil

GOTTINGAE \\
TYPIS DIETERICHIANIS \\
MDCCCXXVIII
\end{center}
\cleardoublepage
\PageSep{2}
%[Blank page]
\PageSep{3}


\PaperTitle{\LARGE GENERAL INVESTIGATIONS \\
{\small OF} \\
CURVED SURFACES \\
{\small BY} \\
{\large KARL FRIEDRICH GAUSS} \\
{\footnotesize PRESENTED TO THE ROYAL SOCIETY, OCTOBER~8, 1827}}


\Article{1.}
Investigations, in which the directions of various straight lines in space are to be
considered, attain a high degree of clearness and simplicity if we employ, as an auxiliary,
a sphere of unit radius described about an arbitrary centre, and suppose the
different points of the sphere to represent the directions of straight lines parallel to
the radii ending at these points. As the position of every point in space is determined
by three coordinates, that is to say, the distances of the point from three mutually
perpendicular fixed planes, it is necessary to consider, first of all, the directions of the
axes perpendicular to these planes. The points on the sphere, which represent these
directions, we shall denote by $(1)$,~$(2)$,~$(3)$. The distance of any one of these points
from either of the other two will be a quadrant; and we shall suppose that the directions
of the axes are those in which the corresponding coordinates increase.


\Article{2.}
It will be advantageous to bring together here some propositions which are frequently
used in questions of this kind.

\Par{I.} The angle between two intersecting straight lines is measured by the arc
between the points on the sphere which correspond to the directions of the lines.

\Par{II.}\Note{1} The orientation of any plane whatever can be represented by the great circle
on the sphere, the plane of which is parallel to the given plane.
\PageSep{4}

\Par{III.} The angle between two planes is equal to the spherical angle between the
great circles representing them, and, consequently, is also measured by the arc intercepted
between the poles of these great circles. And, in like manner, the angle of inclination
of a straight line to a plane is measured by the arc drawn from the point which
corresponds to the direction of the line, perpendicular to the great circle which represents
the orientation of the plane.

\Par{IV.} Letting $x$,~$y$,~$z$; $x'$,~$y'$,~$z'$ denote the coordinates of two points, $r$~the distance
between them, and $L$~the point on the sphere which represents the direction of the line
drawn from the first point to the second, we shall have
\begin{align*}
x' &= x + r \cos(1)L\Add{,} \\
y' &= y + r \cos(2)L\Add{,} \\
z' &= z + r \cos(3)L\Add{.}
\end{align*}

\Par{V.} From this it follows at once that, generally,\Note{2}
\[
\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1\Add{,}
\]
and also, if $L'$~denote any other point on the sphere,
\[
\cos(1)L·\cos(1)L' + \cos(2)L·\cos(2)L' + \cos(3)L·\cos(3)L'
  = \cos LL'.
\]

\Par{VI.}\Note{4} \begin{Theorem}
If $L$, $L'$, $L''$, $L'''$ denote four points on the sphere, and $A$~the angle
which the arcs $LL'$, $L''L'''$ make at their point of intersection, then we shall have
\[
\cos LL''·\cos L'L''' - \cos LL'''·\cos L'L''
  = \sin LL'·\sin L''L'''·\cos A\Add{.}
\]
\end{Theorem}

\textit{Demonstration.} Let $A$ denote also the point of intersection itself, and set
\[
AL = t,\quad AL' = t',\quad AL'' = t'',\quad AL''' = t'''\Add{.}
\]
Then we shall have
\begin{alignat*}{7}
&\cos L L''  &&= \cos t\Chg{·}{}&& \cos t''  &&+ \sin t  && \sin t''  && \cos A\Add{,} \\
&\cos L'L''' &&= \cos t' && \cos t''' &&+ \sin t' && \sin t''' && \cos A\Add{,} \\
&\cos L L''' &&= \cos t  && \cos t''' &&+ \sin t  && \sin t''' && \cos A\Add{,} \\
&\cos L'L''  &&= \cos t' && \cos t''  &&+ \sin t' && \sin t''  && \cos A\Add{;}
\end{alignat*}
and consequently,\Note{3}
\begin{multline*}
\cos LL''·\cos L'L''' - \cos LL'''·\cos L'L'' \\
\begin{aligned}
&= \cos A (\cos t  \cos t''  \sin t' \sin t'''
         + \cos t' \cos t''' \sin t  \sin t'' \\
&\qquad  - \cos t  \cos t''' \sin t' \sin t''
         - \cos t' \cos t''  \sin t  \sin t''') \\
&= \cos A (\cos t   \sin t'   - \sin t   \cos t')
          (\cos t'' \sin t''' - \sin t'' \cos t''')\NoteMark \\
&= \cos A·\sin (t' - t)·\sin (t''' - t'') \\
&= \cos A·\sin LL'·\sin L''L'''\Add{.}
\end{aligned}
\end{multline*}
\PageSep{5}

But as there are for each great circle two branches going out from the point~$A$,
these two branches form at this point two angles whose sum is~$180°$. But our analysis
shows that those branches are to be taken whose directions are in the sense from the
point $L$~to~$L'$, and from the point $L''$~to~$L'''$; and since great circles intersect in two
points, it is clear that either of the two points can be chosen arbitrarily. Also, instead
of the angle~$A$, we can take the arc between the poles of the great circles of which the
arcs $LL'$,~$L''L'''$ are parts. But it is evident that those poles are to be chosen which
are similarly placed with respect to these arcs; that is to say, when we go from $L$~to~$L'$
and from $L''$~to~$L'''$, both of the two poles are to be on the right, or both on the left.

\Par{VII.} Let $L$,~$L'$,~$L''$ be the three points on the sphere and set, for brevity,
\begin{alignat*}{6}
&\cos (1)L   &&= x,\quad&& \cos (2)L   &&= y,\quad&& \cos (3)L   &&= z\Add{,} \\
&\cos (1)L'  &&= x',    && \cos (2)L'  &&= y',    && \cos (3)L'  &&= z'\Add{,} \\
&\cos (1)L'' &&= x'',   && \cos (2)L'' &&= y'',   && \cos (3)L'' &&= z''\Add{;} \\
\end{alignat*}
and also
\[
x y' z'' + x' y'' z + x'' y z' - x y'' z' - x' y z'' - x'' y' z = \Delta\Add{.}
\]
Let $\lambda$~denote the pole of the great circle of which $LL'$~is a part, this pole being the one
that is placed in the same position with respect to this arc as the point~$(1)$ is with
respect to the arc~$(2)(3)$. Then we shall have, by the preceding theorem,
\[
y z' - y' z = \cos (1)\lambda·\sin (2)(3)·\sin LL',
\]
or, because $(2)(3) = 90°$,
\begin{align*}
y z' - y' z &= \cos (1)\lambda·\sin LL', \\
\intertext{and similarly,}
z x' - z' x &= \cos (2)\lambda·\sin LL'\Add{,} \\
x y' - x' y &= \cos (3)\lambda·\sin LL'\Add{.}
\end{align*}
Multiplying these equations by $x''$,~$y''$,~$z''$ respectively, and adding, we obtain, by means
of the second of the theorems deduced in~V,
\[
\Delta = \cos \lambda L''·\sin LL'\Add{.}
\]
Now there are three cases to be distinguished. \emph{First}, when $L''$~lies on the great circle
of which the arc~$LL'$ is a part, we shall have $\lambda L'' = 90°$, and consequently, $\Delta = 0$.
If $L''$~does not lie on that great circle, the \emph{second} case will be when $L''$~is on the same
side as~$\lambda$; the \emph{third} case when they are on opposite sides. In the last two cases the
points $L$,~$L'$,~$L''$ will form a spherical triangle, and in the second case these points will lie
in the same order as the points $(1)$,~$(2)$,~$(3)$, and in the opposite order in the third case.
\PageSep{6}
Denoting the angles of this triangle simply by $L$,~$L'$,~$L''$ and the perpendicular drawn on
the sphere from the point~$L''$ to the side~$LL'$ by~$p$, we shall have
\[
\sin p = \sin L·\sin LL'' = \sin L'·\sin L' L'',
\]
and
\[
\lambda L'' = 90° \mp p,
\]
the upper sign being taken for the second case, the lower for the third. From this
it follows that
\begin{align*}
±\Delta &= \sin L·\sin LL'·\sin LL''
   = \sin L'·\sin LL'·\sin L'L'' \\
  &= \sin L''·\sin LL''·\sin L'L''\Add{.}
\end{align*}
Moreover, it is evident that the first case can be regarded as contained in the second or
third, and it is easily seen that the expression~$±\Delta$ represents six times the volume of
the pyramid formed by the points $L$,~$L'$,~$L''$ and the centre of the sphere. Whence,
finally, it is clear that the expression~$±\frac{1}{6}\Delta$ expresses generally the volume of any
pyramid contained between the origin of coordinates and the three points whose coordinates
are $\Typo{z}{x}$,~$y$,~$z$; $x'$,~$y'$,~$z'$; $x''$,~$y''$,~$z''$.\Note{5}


\Article{3.}

A curved surface is said to possess continuous curvature at one of its points~$A$, if the
directions of all the straight lines drawn from $A$ to points of the surface at an infinitely
small distance from~$A$ are deflected infinitely little from one and the same plane passing
through~$A$. This plane is said to \emph{touch} the surface at the point~$A$. If this condition is
not satisfied for any point, the continuity of the curvature is here interrupted, as happens,
for example, at the vertex of a cone. The following investigations will be restricted to
such surfaces, or to such parts of surfaces, as have the continuity of their curvature
nowhere interrupted. We shall only observe now that the methods used to determine
the position of the tangent plane lose their meaning at singular points, in which the
continuity of the curvature is interrupted, and must lead to indeterminate solutions.


\Article{4.}

The orientation of the tangent plane is most conveniently studied by means of the
direction of the straight line normal to the plane at the point~$A$, which is also called the
normal to the curved surface at the point~$A$. We shall represent the direction of this
normal by the point~$L$ on the auxiliary sphere, and we shall set
\[
\cos (1)L = X,\quad \cos (2)L = Y,\quad \cos (3)L = Z;
\]
and denote the coordinates of the point~$A$ by $x$,~$y$,~$z$. Also let $x + dx$, $y + dy$, $z + dz$
be the coordinates of another point~$A'$ on the curved surface; $ds$~its distance from~$A$,
\PageSep{7}
which is infinitely small; and finally, let $\lambda$ be the point on the sphere representing the
direction of the element~$AA'$. Then we shall have
\[
dx = ds·\cos (1)\lambda,\quad
dy = ds·\cos (2)\lambda,\quad
dz = ds·\cos (3)\lambda
\]
and, since $\lambda L$~must be equal to~$90°$,
\[
X\cos (1)\lambda + Y\cos (2)\lambda + Z\cos (3)\lambda = 0\Add{.}
\]
By combining these equations we obtain
\[
X\, dx + Y\, dy + Z\, dz = 0.
\]

There are two general methods for defining the nature of a curved surface. The
\emph{first} uses the equation between the coordinates $x$,~$y$,~$z$, which we may suppose reduced to
the form $W = 0$, where $W$~will be a function of the indeterminates $x$,~$y$,~$z$. Let the complete
differential of the function~$W$ be
\[
dW = P\, dx + Q\, dy + R\, dz
\]
and on the curved surface we shall have
\[
P\, dx + Q\, dy + R\, dz = 0\Add{,}
\]
and consequently,
\[
P \cos (1)\lambda + Q \cos (2)\lambda + R \cos (3)\lambda = 0\Add{.}
\]
Since this equation, as well as the one we have established above, must be true for the
directions of all elements~$ds$ on the curved surface, we easily see that $X$,~$Y$,~$Z$ must be
proportional to $P$,~$Q$,~$R$ respectively, and consequently, since\Note{6}
\[
X^{2} + Y^{2} + Z^{2} = 1,\NoteMark
\]
we shall have either
\begin{align*}
X &= \frac{P}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
Y &= \frac{Q}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
Z &= \frac{R}{\Sqrt{P^{2} + Q^{2} + R^{2}}} \\
\intertext{or}
X &= \frac{-P}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
Y &= \frac{-Q}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
Z &= \frac{-R}{\Sqrt{P^{2} + Q^{2} + R^{2}}}\Add{.}
\end{align*}

The \emph{second}\Note{7} method expresses the coordinates in the form of functions of two variables,
$p$,~$q$. Suppose that differentiation of these functions gives
\begin{alignat*}{2}
dx &= a\, dp &&+ a'\, dq\Add{,} \\
dy &= b\, dp &&+ b'\, dq\Add{,} \\
dz &= c\, dp &&+ c'\, dq\Add{.}
\end{alignat*}
\PageSep{8}
Substituting these values in the formula given above, we obtain
\[
(aX + bY + cZ)\, dp + (a'X + b'Y + c'Z)\, dq = 0\Add{.}
\]
Since this equation must hold independently of the values of the differentials $dp$,~$dq$,
we evidently shall have
\[
aX + bY + cZ = 0,\quad a'X + b'Y + c'Z = 0\Add{.}
\]
From this we see that $X$,~$Y$,~$Z$ will be proportioned to the quantities
\[
bc' - cb',\quad ca' - ac',\quad ab' - ba'\Add{.}
\]
Hence, on setting, for brevity,
\[
\Sqrt{(bc' - cb')^{2} + (ca' - ac')^{2} + (ab' - ba')^{2}} = \Delta\Add{,}
\]
we shall have either
\begin{align*}
X &= \frac{bc' - cb'}{\Delta},\quad&
Y &= \frac{ca' - ac'}{\Delta},\quad&
Z &= \frac{ab' - ba'}{\Delta}
\intertext{or}
X &= \frac{cb' - bc'}{\Delta},\quad&
Y &= \frac{ac' - ca'}{\Delta},\quad&
Z &= \frac{ba' - ab'}{\Delta}\Add{.}
\end{align*}

With these two general methods is associated a \emph{third}, in which one of the coordinates,
$z$,~say, is expressed in the form of a function of the other two, $x$,~$y$. This method is
evidently only a particular case either of the first method, or of the second. If we set
\[
dz = t\, dx + u\, dy
\]
we shall have either
\begin{align*}
X &= \frac{-t}{\Sqrt{1 + t^{2} + u^{2}}}, &
Y &= \frac{-u}{\Sqrt{1 + t^{2} + u^{2}}}, &
Z &= \frac{ 1}{\Sqrt{1 + t^{2} + u^{2}}} \\
\intertext{or}
X &= \frac{ t}{\Sqrt{1 + t^{2} + u^{2}}}, &
Y &= \frac{ u}{\Sqrt{1 + t^{2} + u^{2}}}, &
Z &= \frac{-1}{\Sqrt{1 + t^{2} + u^{2}}}\Add{.}
\end{align*}


\Article{5.}

The two solutions found in the preceding article evidently refer to opposite points of
the sphere, or to opposite directions, as one would expect, since the normal may be drawn
toward either of the two sides of the curved surface. If we wish to distinguish between
the two regions bordering upon the surface, and call one the exterior region and the other
the interior region, we can then assign to each of the two normals its appropriate solution
by aid of the theorem derived in \Art{2}~(VII), and at the same time establish a criterion
for distinguishing the one region from the other.
\PageSep{9}

In the first method, such a criterion is to be drawn from the sign of the quantity~$W$.
Indeed, generally speaking, the curved surface divides those regions of space in which $W$
keeps a positive value from those in which the value of~$W$ becomes negative. In fact, it
is easily seen from this theorem that, if $W$ takes a positive value toward the exterior
region, and if the normal is supposed to be drawn outwardly, the first solution is to be
taken. Moreover, it will be easy to decide in any case whether the same rule for the
sign of~$W$ is to hold throughout the entire surface, or whether for different parts there
will be different rules. As long as the coefficients $P$,~$Q$,~$R$ have finite values and do not
all vanish at the same time, the law of continuity will prevent any change.

If we follow the second method, we can imagine two systems of curved lines on the
curved surface, one system for which $p$~is variable, $q$~constant; the other for which $q$~is
variable, $p$~constant. The respective positions of these lines with reference to the exterior
region will decide which of the two solutions must be taken. In fact, whenever
the three lines, namely, the branch of the line of the former system going out from the
point~$A$ as $p$~increases, the branch of the line of the latter system going out from the point
$A$ as $q$~increases, and the normal drawn toward the exterior region, are \emph{similarly} placed as
the $x$,~$y$,~$z$ axes respectively from the origin of abscissas (\eg, if, both for the former
three lines and for the latter three, we can conceive the first directed to the left, the
second to the right, and the third upward), the first solution is to be taken. But whenever
the relative position of the three lines is opposite to the relative position of the
$x$,~$y$,~$z$ axes, the second solution will hold.

In the third method, it is to be seen whether, when $z$~receives a positive increment, $x$~and~$y$
remaining constant, the point crosses toward the exterior or the interior region.
In the former case, for the normal drawn outward, the first solution holds; in the latter
case, the second.


\Article{6.}

Just as each definite point on the curved surface is made to correspond to a definite
point on the sphere, by the direction of the normal to the curved surface which is transferred
to the surface of the sphere, so also any line whatever, or any figure whatever, on
the latter will be represented by a corresponding line or figure on the former. In the
comparison of two figures corresponding to one another in this way, one of which will be
as the map of the other, two important points are to be considered, one when quantity
alone is considered, the other when, disregarding quantitative relations, position alone
is considered.

The first of these important points will be the basis of some ideas which it seems
judicious to introduce into the theory of curved surfaces. Thus, to each part of a curved
\PageSep{10}
surface inclosed within definite limits we assign a \emph{total} or \emph{integral curvature}, which is
represented by the area of the figure on the sphere corresponding to it. From this
integral curvature must be distinguished the somewhat more specific curvature which we
shall call the\Note{8} \emph{measure of curvature}. The latter refers to a \emph{point} of the surface, and shall
denote the quotient obtained when the integral curvature of the surface element about
a point is divided by the area of the element itself; and hence it denotes the ratio of the
infinitely small areas which correspond to one another on the curved surface and on the
sphere. The use of these innovations will be abundantly justified, as we hope, by what
we shall explain below. As for the terminology, we have thought it especially desirable
that all ambiguity be avoided. For this reason we have not thought it advantageous to
follow strictly the analogy of the terminology commonly adopted (though not approved by
all) in the theory of plane curves, according to which the measure of curvature should be
called simply curvature, but the total curvature, the amplitude. But why not be free in
the choice of words, provided they are not meaningless and not liable to a misleading
interpretation?

The position of a figure on the sphere can be either similar to the position of the
corresponding figure on the curved surface, or opposite (inverse). The former is the case
when two lines going out on the curved surface from the same point in different, but not
opposite directions, are represented on the sphere by lines similarly placed, that is, when
the map of the line to the right is also to the right; the latter is the case when the contrary
holds. We shall distinguish these two cases by the positive or negative \emph{sign} of the
measure of curvature. But evidently this distinction can hold only when on each surface
we choose a definite face on which we suppose the figure to lie. On the auxiliary sphere
we shall use always the exterior face, that is, that turned away from the centre; on the
curved surface also there may be taken for the exterior face the one already considered,
or rather that face from which the normal is supposed to be drawn. For, evidently, there
is no change in regard to the similitude of the figures, if on the curved surface both the
figure and the normal be transferred to the opposite side, so long as the image itself
is represented on the same side of the sphere.

The positive or negative sign, which we assign to the \emph{measure} of curvature according
to the position of the infinitely small figure, we extend also to the integral curvature
of a finite figure on the curved surface. However, if we wish to discuss the general case,
some explanations will be necessary, which we can only touch here briefly. So long
as the figure on the curved surface is such that to \emph{distinct} points on itself there correspond
distinct points on the sphere, the definition needs no further explanation. But
whenever this condition is not satisfied, it will be necessary to take into account twice
or several times certain parts of the figure on the sphere. Whence for a similar, or
\PageSep{11}
inverse position, may arise an accumulation of areas, or the areas may partially or
wholly destroy each other. In such a case, the simplest way is to suppose the curved
surface divided into parts, such that each part, considered separately, satisfies the above
condition; to assign to each of the parts its integral curvature, determining this magnitude
by the area of the corresponding figure on the sphere, and the sign by the position
of this figure; and, finally, to assign to the total figure the integral curvature
arising from the addition of the integral curvatures which correspond to the single parts.
So, generally, the integral curvature of a figure is equal to $\int k\, d\sigma$, $d\sigma$~denoting the
element of area of the figure, and $k$~the measure of curvature at any point. The principal
points concerning the geometric representation of this integral reduce to the following.
To the perimeter of the figure on the curved surface (under the restriction
of \Art{3}) will correspond always a closed line on the sphere. If the latter nowhere
intersect itself, it will divide the whole surface of the sphere into two parts, one of
which will correspond to the figure on the curved surface; and its area (taken as
positive or negative according as, with respect to its perimeter, its position is similar,
or inverse, to the position of the figure on the curved surface) will represent the integral
curvature of the figure on the curved surface. But whenever this line intersects
itself once or several times, it will give a complicated figure, to which, however, it is
possible to assign a definite area as legitimately as in the case of a figure without
nodes; and this area, properly interpreted, will give always an exact value for the
integral curvature. However, we must reserve for another occasion\Note{9} the more extended
exposition of the theory of these figures viewed from this very general standpoint.


\Article{7.}

We shall now find a formula which will express the measure of curvature for
any point of a curved surface. Let $d\sigma$~denote the area of an element of this surface;
then $Z\, d\sigma$~will be the area of the projection of this element on the plane of the coordinates
$x$,~$y$; and consequently, if $d\Sigma$~is the area of the corresponding element on the
sphere, $Z\, d\Sigma$~will be the area of its projection on the same plane. The positive or
negative sign of~$Z$ will, in fact, indicate that the position of the projection is similar or
inverse to that of the projected element. Evidently these projections have the same
ratio as to quantity and the same relation as to position as the elements themselves.
Let us consider now a triangular element on the curved surface, and let us suppose
that the coordinates of the three points which form its projection are
\begin{alignat*}{3}
&x,           && y\Add{,} \\
&x + dx,\quad && y + dy\Add{,} \\
&x + \delta x,\quad && y + \delta y\Add{.}
\end{alignat*}
\PageSep{12}
The double area of this triangle will be expressed by the formula
\[
dx·\delta y - dy·\delta x\Add{,}
\]
and this will be in a positive or negative form according as the position of the side
from the first point to the third, with respect to the side from the first point to the
second, is similar or opposite to the position of the $y$-axis of coordinates with respect
to the $x$-axis of coordinates.

In like manner, if the coordinates of the three points which form the projection of
the corresponding element on the sphere, from the centre of the sphere as origin, are
\begin{alignat*}{3}
&X,           && Y\Add{,} \\
&X + dX,\quad && Y + dY\Add{,} \\
&X + \delta X,\quad && Y + \delta Y\Add{,}
\end{alignat*}
the double area of this projection will be expressed by
\[
dX·\delta Y - dY·\delta X\Add{,}
\]
and the sign of this expression is determined in the same manner as above. Wherefore
the measure of curvature at this point of the curved surface will be
\[
k = \frac{dX·\delta Y - dY·\delta X}{dx·\delta y - dy·\delta x}\Add{.}
\]
If now we suppose the nature of the curved surface to be defined according to the third
method considered in \Art{4}, $X$~and~$Y$ will be in the form of functions of the quantities
$x$,~$y$. We shall have, therefore,\Note{10}
\begin{alignat*}{2}
dX &= \frac{\dd X}{\dd x}\, dx &&+ \frac{\dd X}{\dd y}\, dy\Add{,} \\
\delta X &= \frac{\dd X}{\dd x}\, \delta x
          &&+ \frac{\dd X}{\dd y}\, \delta y\Add{,} \\
dY &= \frac{\dd Y}{\dd x}\, dx &&+ \frac{\dd Y}{\dd y}\, dy\Add{,} \\
\delta Y &= \frac{\dd Y}{\dd x}\, \delta x
          &&+ \frac{\dd Y}{\dd y}\, \delta y\Add{.}
\end{alignat*}
When these values have been substituted, the above expression becomes
\[
k = \frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
  - \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x}\Add{.}
\]
\PageSep{13}
Setting, as above,
\[
\frac{\dd z}{\dd x} = t,\quad \frac{\dd z}{\dd y} = u
\]
and also
\[
\frac{\dd^{2} z}{\dd x^{2}} = T,\quad
\frac{\dd^{2} z}{\dd x·\dd y} = U,\quad
\frac{\dd^{2} z}{\dd y^{2}} = V\Add{,}
\]
or
\[
dt = T\, dx + U\, dy,\quad
du = U\, dx + V\, dy\Add{,}
\]
we have from the formulæ given above
\[
X = -tZ,\quad Y = -uZ,\quad (1 - t^{2} - u^{2})Z^{2} = 1\Add{;}
\]
and hence
\begin{gather*}
\begin{alignedat}{2}
dX &= -Z\, dt &&- t\, dZ\Add{,} \\
dY &= -Z\, du &&- u\, dZ\Add{,}
\end{alignedat} \\
(1 + t^{2} + u^{2})\, dZ + Z(t\, dt + u\, du) = 0\Add{;}
\end{gather*}
or\Note{11}
\begin{align*}
dZ &= -Z^{3}(t\, dt + u\, du)\Add{,} \\
dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3} tu\, du\Add{,} \\
dY &= +Z^{3}tu\, dt - Z^{3}(1 + t^{2})\, du\Add{;}\NoteMark
\end{align*}
and so
\begin{align*}
\frac{\dd X}{\dd x} &= Z^{3}\bigl(-(1 + u^{2})T + tuU\bigr)\Add{,} \\
\frac{\dd X}{\dd y} &= Z^{3}\bigl(-(1 + u^{2})U + tuV\bigr)\Add{,} \\
\frac{\dd Y}{\dd x} &= Z^{3}\bigl( tuT - (1 + t^{2})U\bigr)\Add{,} \\
\frac{\dd Y}{\dd y} &= Z^{3}\bigl( tuU - (1 + t^{2})V\bigr)\Add{.}
\end{align*}
Substituting these values in the above expression, it becomes
\begin{align*}
k &= Z^{6}(TV - U^{2}) (1 + t^{2} + u^{2}) = Z^{4} (TV - U^{2}) \\
  &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}\Add{.}
\end{align*}


\Article{8.}

By a suitable choice of origin and axes of coordinates, we can easily make the
values of the quantities $t$,~$u$,~$U$ vanish for a definite point~$A$. Indeed, the first two
\PageSep{14}
conditions will be fulfilled at once if the tangent plane at this point be taken for the
$xy$-plane. If, further, the origin is placed at the point $A$~itself, the expression for
the coordinate~$z$ evidently takes the form
\[
z = \tfrac{1}{2}T°x^{2} + U°xy + \tfrac{1}{2}V°y^{2} + \Omega\Add{,}
\]
where $\Omega$~will be of higher degree than the second. Turning now the axes of $x$~and~$y$
through an angle~$M$ such that
\[
\tan 2M = \frac{2U°}{T° - V°}\Add{,}
\]
it is easily seen that there must result an equation of the form
\[
z = \tfrac{1}{2}Tx^{2} + \tfrac{1}{2}Vy^{2} + \Omega\Add{.}
\]
In this way the third condition is also satisfied. When this has been done, it is evident
that

\Par{I.} If the curved surface be cut by a plane passing through the normal itself and
through the $x$-axis, a plane curve will be obtained, the radius of curvature of which
at the point~$A$ will be equal to~$\dfrac{1}{T}$, the positive or negative sign indicating that the
curve is concave or convex toward that region toward which the coordinates~$z$ are
positive.

\Par{II.} In like manner $\dfrac{1}{V}$~will be the radius of curvature at the point~$A$ of the plane
curve which is the intersection of the surface and the plane through the $y$-axis and
the $z$-axis.

\Par{III.} Setting $z = r \cos\phi$, $y = r \sin \phi$, the equation becomes
\[
z = \tfrac{1}{2}(T\cos^{2}\phi + V\sin^{2}\phi) r^{2} + \Omega\Add{,}
\]
from which we see that if the section is made by a plane through the normal at~$A$
and making an angle~$\phi$ with the $x$-axis, we shall have a plane curve whose radius of
curvature at the point~$A$ will be
\[
\frac{1}{T\cos^{2}\phi + V\sin^{2}\phi}\Add{.}
\]

\Par{IV.} Therefore, whenever we have $T = V$, the radii of curvature in \emph{all} the normal
planes will be equal. But if $T$~and~$V$ are not equal, it is evident that, since for any
value whatever of the angle~$\phi$, $T\cos^{2}\phi + V\sin^{2}\phi$ falls between $T$~and~$V$, the radii of
curvature in the principal sections considered in I.~and~II. refer to the extreme curvatures;
that is to say, the one to the maximum curvature, the other to the minimum,
\PageSep{15}
if $T$~and~$V$ have the same sign. On the other hand, one has the greatest convex
curvature, the other the greatest concave curvature, if $T$~and~$V$ have opposite signs.
These conclusions contain almost all that the illustrious Euler\Note{12} was the first to prove
on the curvature of curved surfaces.

\Par{V.} The measure of curvature at the point~$A$ on the curved surface takes the
very simple form
\[
k = TV,
\]
whence we have the

\begin{Theorem}
The measure of curvature at any point whatever of the surface is equal to a
fraction whose numerator is unity, and whose denominator is the product of the two extreme
radii of curvature of the sections by normal planes.\
\end{Theorem}

At the same time it is clear that the measure of curvature is positive for concavo-concave
or convexo-convex surfaces (which distinction is not essential), but negative
for concavo-convex surfaces. If the surface consists of parts of each kind, then
on the lines separating the two kinds the measure of curvature ought to vanish. Later
we shall make a detailed study of the nature of curved surfaces for which the measure
of curvature everywhere vanishes.


\Article{9.}

The general formula for the measure of curvature given at the end of \Art{7} is
the most simple of all, since it involves only five elements. We shall arrive at a
more complicated formula, indeed, one involving nine elements, if we wish to use the
first method of representing a curved surface. Keeping the notation of \Art{4}, let us
set also
\begin{align*}
\frac{\dd^{2} W}{\dd x^{2}} &= P', &
\frac{\dd^{2} W}{\dd y^{2}} &= Q', &
\frac{\dd^{2} W}{\dd z^{2}} &= R'\Add{,} \\
\frac{\dd^{2} W}{\dd y·\dd z} &= P'', &
\frac{\dd^{2} W}{\dd x·\dd z} &= Q'', &
\frac{\dd^{2} W}{\dd x·\dd y} &= R''\Add{,}
\end{align*}
so that
\begin{alignat*}{4}
dP &= P'\,  &&dx + R''\, &&dy + Q''\, &&dz\Add{,} \\
dQ &= R''\, &&dx + Q' \, &&dy + P''\, &&dz\Add{,} \\
dR &= Q''\, &&dx + P''\, &&dy + R' \, &&dz\Add{.}
\end{alignat*}
Now since $t = -\dfrac{P}{R}$, we find through differentiation
\[
R^{2}\, dt = -R\, dP + P\, dR
  = (PQ'' - RP')\, dx + (PP'' - RR'')\, dy + (PR' - RQ'')\, dz\Add{,}
\]
\PageSep{16}
or, eliminating~$dz$ by means of the equation
\begin{gather*}
P\, dx + Q\, dy + R\, dz = 0, \\
R^{3}\, dt
  = (-R^{2}P' + 2PRQ'' - P^{2}R')\, dx + (PRP'' + QRQ'' - PQR' - R^{2}R'')\, dy.
\end{gather*}
In like manner we obtain
\[
R^{3}\, du
  = (PRP'' + QRQ'' - PQR' - R^{2}R'')\, dx + (-R^{2}Q' + 2QRP'' - Q^{2}R')\, dy\Add{.}
\]
From this we conclude that
\begin{align*}
R^{3}T &= -R^{2}P' + 2PRQ'' - P^{2}R'\Add{,} \\
R^{3}U &= PRP'' + QRQ'' - PQR' - R^{2}R''\Add{,} \\
R^{3}V &= -R^{2}Q' + 2QRP'' - Q^{2}R'\Add{.}
\end{align*}
Substituting these values in the formula of \Art{7}, we obtain for the measure of curvature~$k$
the following symmetric expression:
\begin{multline*}
(P^{2} + Q^{2} + R^{2})^{2}k
  = P^{2}(Q'R' - P''^{2})
  + Q^{2}(P'R' - Q''^{2})
  + R^{2}(P'Q' - R''^{2}) \\
  + 2QR(Q''R'' - P'P'')
  + 2PR(P''R'' - Q'Q'')
  + 2PQ(P''Q'' - R'R'')\Add{.}
\end{multline*}


\Article{10.}

We obtain a still more complicated formula, indeed, one involving fifteen elements,
if we follow the second general method of defining the nature of a curved surface. It
is, however, very important that we develop this formula also. Retaining the notations
of \Art{4}, let us put also
\begin{align*}
\frac{\dd^{2}x}{\dd p^{2}} &= \alpha, &
\frac{\dd^{2}x}{\dd p·\dd q} &= \alpha', &
\frac{\dd^{2}x}{\dd q^{2}} &= \alpha''\Add{,} \\
%
\frac{\dd^{2}y}{\dd p^{2}} &= \beta, &
\frac{\dd^{2}y}{\dd p·\dd q} &= \beta', &
\frac{\dd^{2}y}{\dd q^{2}} &= \beta''\Add{,} \\
%
\frac{\dd^{2}z}{\dd p^{2}} &= \gamma, &
\frac{\dd^{2}z}{\dd p·\dd q} &= \gamma', &
\frac{\dd^{2}z}{\dd q^{2}} &= \gamma''\Add{;} \\
\end{align*}
and let us put, for brevity,
\begin{align*}
bc' - cb' &= A\Add{,} \\
ca' - ac' &= B\Add{,} \\
ab' - ba' &= C\Add{.}
\end{align*}
First we see that
\[
A\, dx + B\, dy + C\, dz = 0,
\]
or
\[
dz = -\frac{A}{C}\, dx - \frac{B}{C}\, dy.
\]
\PageSep{17}
Thus, inasmuch as $z$~may be regarded as a function of $x$,~$y$, we have
\begin{align*}
\frac{\dd z}{\dd x} &= t = -\frac{A}{C}\Add{,} \\
\frac{\dd z}{\dd y} &= u = -\frac{B}{C}\Add{.}
\end{align*}
Then from the formulæ
\[
dx = a\, dp + a'\, dq,\quad
dy = b\, dp + b'\, dq,
\]
we have
\begin{alignat*}{4}
&C\, dp = &&b'\, &&dx - a'\, &&dy\Add{,} \\
&C\, dq =-&&b\,  &&dx + a\,  &&dy\Add{.}
\end{alignat*}
Thence we obtain for the total differentials of $t$,~$u$
\begin{alignat*}{2}
C^{3}\, dt
  &= \left(A\, \frac{\dd C}{\dd p} - C\, \frac{\dd A}{\dd p}\right)(b'\, dx - a'\, dy)
   + \left(C\, \frac{\dd A}{\dd q} - A\, \frac{\dd C}{\dd q}\right)(b \, dx - a \, dy)\Add{,} \\
%
C^{3}\, du
  &= \left(B\, \frac{\dd C}{\dd p} - C\, \frac{\dd B}{\dd p}\right)(b'\, dx - a'\, dy)
   + \left(C\, \frac{\dd B}{\dd q} - B\, \frac{\dd C}{\dd q}\right)(b \, dx - a \, dy)\Add{.}
\end{alignat*}
If now we substitute in these formulæ
\begin{alignat*}{4}
\frac{\dd A}{\dd p} &= c'\beta  &&+ b\gamma'  &&- c\beta'  &&- b'\gamma\Add{,} \\
\frac{\dd A}{\dd q} &= c'\beta' &&+ b\gamma'' &&- c\beta'' &&- b'\gamma'\Add{,} \\
%
\frac{\dd B}{\dd p} &= a'\gamma  &&+ c\alpha'  &&- a\gamma'  &&- c'\alpha\Add{,} \\
\frac{\dd B}{\dd q} &= a'\gamma' &&+ c\alpha'' &&- a\gamma'' &&- c'\alpha'\Add{,} \\
%
\frac{\dd C}{\dd p} &= b'\alpha  &&+ a\beta'  &&- b\alpha'  &&- a'\beta\Add{,} \\
\frac{\dd C}{\dd q} &= b'\alpha' &&+ a\beta'' &&- b\alpha'' &&- a'\beta'\Add{;}
\end{alignat*}
and if we note that the values of the differentials $dt$,~$du$ thus obtained must be equal,
independently of the differentials $dx$,~$dy$, to the quantities $T\, dx + U\, dy$, $U\, dx + V\, dy$
respectively, we shall find, after some sufficiently obvious transformations,
\begin{align*}
C^{3}T &= \alpha Ab'^{2} + \beta Bb'^{2} + \gamma Cb'^{2} \\
&\quad- 2\alpha' Abb' - 2\beta' Bbb' - 2\gamma' Cbb' \\
&\quad+ \alpha'' Ab^{2} + \beta'' Bb^{2} + \gamma'' Cb^{2}\Add{,} \\
\PageSep{18}
C^{3}U &= -\alpha Aa'b' - \beta Ba'b' - \gamma Ca'b' \\
&\quad+ \alpha' A(ab' + ba') + \beta' B(ab' + ba') + \gamma' C(ab' + ba') \\
&\quad- \alpha'' Aab - \beta'' Bab - \gamma'' Cab\Add{,} \\
%
C^{3}V &= \alpha Aa'^{2} + \beta Ba'^{2} + \gamma Ca'^{2} \\
&\quad- 2\alpha' Aaa' - 2\beta' Baa' - 2\gamma' Caa' \\
&\quad+ \alpha'' Aa^{2} + \beta'' Ba^{2} + \gamma'' Ca^{2}\Add{.}
\end{align*}
Hence, if we put, for the sake of brevity,\Note{13}
\begin{alignat*}{4}
&A\alpha   &&+ B\beta   &&+ C\gamma   &&= D\Add{,}\NoteMark
\Tag{(1)} \\
&A\alpha'  &&+ B\beta'  &&+ C\gamma'  &&= D'\Add{,}
\Tag{(2)} \\
&A\alpha'' &&+ B\beta'' &&+ C\gamma'' &&= D''\Add{,}
\Tag{(3)}
\end{alignat*}
we shall have
\begin{align*}
C^{3}T &=  Db'^{2} - 2D'bb' + D'' b^{2}\Add{,} \\
C^{3}U &= -Da'b' + D'(ab' + ba') - D''ab\Add{,} \\
C^{3}V &=  Da'^{2} - 2D'aa' + D''a^{2}\Add{.}
\end{align*}
From this we find, after the reckoning has been carried out,
\[
C^{6}(TV - U^{2}) = (DD'' - D'^{2}) (ab' - ba')^{2} = (DD'' - D'^{2}) C^{2}\Add{,}
\]
and therefore the formula for the measure of curvature
\[
k = \frac{DD'' - D'^{2}}{(A^{2} + B^{2} + C^{2})^{2}}\Add{.}
\]


\Article{11.}

By means of the formula just found we are going to establish another, which may
be counted among the most productive theorems in the theory of curved surfaces.
Let us introduce the following notation:
\begin{alignat*}{4}
&a^{2} &&+ b^{2} &&+ c^{2} &&= E\Add{,} \\
&aa' &&+ bb' &&+ cc' &&= F\Add{,} \\
&a'^{2} &&+ b'^{2} &&+ c'^{2} &&= G\Add{;}
\end{alignat*}
\begin{alignat*}{7}
&a &&\alpha  &&+b &&\beta  &&+c &&\gamma  &&= m  \Add{,}
\Tag{(4)} \\
&a &&\alpha' &&+b &&\beta' &&+c &&\gamma' &&= m' \Add{,}
\Tag{(5)} \\
&a &&\alpha''&&+b &&\beta''&&+c &&\gamma''&&= m''\Add{;}
\Tag{(6)} \displaybreak[1] \\
%
&a'\,&&\alpha  &&+b'\,&&\beta  &&+c'\,&&\gamma  &&= n  \Add{,}
\Tag{(7)} \\
&a'&&\alpha' &&+b'&&\beta' &&+c'&&\gamma' &&= n' \Add{,}
\Tag{(8)} \\
&a'&&\alpha''&&+b'&&\beta''&&+c'&&\gamma''&&= n''\Add{;}
\Tag{(9)}
\end{alignat*}
\[
A^{2} + B^{2} + C^{2} = EG - F^{2} = \Delta\Add{.}
\]
\PageSep{19}

%[** TN: Added parentheses around equation numbers]
Let us eliminate from the equations (1),~(4),~(7) the quantities $\beta$,~$\gamma$, which is done by
multiplying them by $bc' - cb'$, $b'C - c'B$, $cB - bC$ respectively and adding. In this
way we obtain
\begin{multline*}
\bigl(A(bc' - cb') + a(b'C - c'B) + a'(cB - bC)\bigr)\alpha \\
  = D(bc' - cb') + m(b'C - c'B) + n(cB - bC)\Add{,}
\end{multline*}
an equation which is easily transformed into
\[
AD = \alpha\Delta + a(nF - mG) + a'(mF - nE)\Add{.}
\]
Likewise the elimination of $\alpha$,~$\gamma$ or $\alpha$,~$\beta$ from the same equations gives
\begin{alignat*}{4}
&BD &&= \beta \Delta &&+ b(nF - mG) &&+ b'(mF - nE)\Add{,} \\
&CD &&= \gamma\Delta &&+ c(nF - mG) &&+ c'(mF - nE)\Add{.}
\end{alignat*}
Multiplying these three equations by $\alpha''$,~$\beta''$,~$\gamma''$ respectively and adding, we obtain
\[
DD'' = (\alpha\alpha''+ \beta\beta'' + \gamma\gamma'')\Delta
  + m''(nF - mG) + n''(mF - nE)\Add{.}
\Tag{(10)}
\]

%[** TN: Added parentheses around equation numbers]
If we treat the equations (2),~(5),~(8) in the same way, we obtain
\begin{alignat*}{4}
&AD' &&= \alpha'\Delta &&+ a (n'F - m'G) &&+ a'(m'F - n'E)\Add{,} \\
&BD' &&= \beta' \Delta &&+ b (n'F - m'G) &&+ b'(m'F - n'E)\Add{,} \\
&CD' &&= \gamma'\Delta &&+ c (n'F - m'G) &&+ c'(m'F - n'E)\Add{;}
\end{alignat*}
and after these equations are multiplied by $\alpha'$,~$\beta'$,~$\gamma'$ respectively, addition gives
\[
D'^{2} = (\alpha'^{2} + \beta'^{2} + \gamma'^{2})\Delta
  + m'(n'F - m'G) + n'(m'F - n'E)\Add{.}
\]

A combination of this equation with equation~(10) gives
\begin{multline*}
DD'' - D'^{2} = (\alpha\alpha'' + \beta\beta'' + \gamma\gamma''
  - \alpha'^{2} - \beta'^{2} - \gamma'^{2})\Delta \\
  + E(n'^{2} - nn'') + F(nm'' - 2m'n' + mn'') + G(m'^{2} - mm'')\Add{.}
\end{multline*}
It is clear that we have
\[
\frac{\dd E}{\dd p} = 2m,\
\frac{\dd E}{\dd q} = 2m',\quad
\frac{\dd F}{\dd p} = m' + n,\
\frac{\dd F}{\dd q} = m'' + n',\quad
\frac{\dd G}{\dd p} = 2n',\
\frac{\dd G}{\dd q} = 2n'',
\]
or\Note{14}
\begin{align*}
m   &= \tfrac{1}{2}\, \frac{\dd E}{\dd p}, &
m'  &= \tfrac{1}{2}\, \frac{\dd E}{\dd q}, &
m'' &= \frac{\dd F}{\dd q} - \tfrac{1}{2}\, \frac{\dd G}{\dd p}\Add{,}\NoteMark \\
%
n   &= \frac{\dd F}{\dd p} - \tfrac{1}{2}\, \frac{\dd E}{\dd q}, &
n'  &= \tfrac{1}{2}\, \frac{\dd G}{\dd p}, &
n'' &= \tfrac{1}{2}\, \frac{\dd G}{\dd q}\Add{.}
\end{align*}
Moreover, it is easily shown that we shall have
\begin{align*}
%[** TN: Aligning on equals sign]
\alpha\alpha'' + \beta\beta'' + \gamma\gamma''
  - \alpha'^{2} - \beta'^{2} - \gamma'^{2}
  &= \frac{\dd n}{\dd q} - \frac{\dd n'}{\dd p}
   = \frac{\dd m''}{\dd p} - \frac{\dd m'}{\dd q} \\
  &= -\tfrac{1}{2}·\frac{\dd^{2}E}{\dd q^{2}}
  + \frac{\dd^{2}F}{\dd p·\dd q}
  - \tfrac{1}{2}·\frac{\dd^{2}G}{\dd p^{2}}\Add{.}
\end{align*}
\PageSep{20}
If we substitute these different expressions in the formula for the measure of curvature
derived at the end of the preceding article, we obtain the following formula, which
involves only the quantities $E$,~$F$,~$G$ and their differential quotients of the first and
second orders:
\begin{multline*}
4(EG - F^{2})k
  = E\left(\frac{\dd E}{\dd q}·\frac{\dd G}{\dd q}
       - 2 \frac{\dd F}{\dd p}·\frac{\dd G}{\dd q}
  + \biggl(\frac{\dd G}{\dd p}\biggr)^{2}\right) \\
  + F\left(\frac{\dd E}{\dd p}·\frac{\dd G}{\dd q}
         - \frac{\dd E}{\dd q}·\frac{\dd G}{\dd p}
       - 2 \frac{\dd E}{\dd q}·\frac{\dd F}{\dd q}
       + 4 \frac{\dd \Erratum{E}{F}}{\dd p}·\frac{\dd F}{\dd q}
       - 2 \frac{\dd F}{\dd p}·\frac{\dd G}{\dd p}\right) \\
  + G\left(\frac{\dd E}{\dd p}·\frac{\dd G}{\dd p}
       - 2 \frac{\dd E}{\dd p}·\frac{\dd F}{\dd q}
  + \biggl(\frac{\dd E}{\dd q}\biggr)^{2}\right)
  - 2(EG - F^{2})\left(
      \frac{\dd^{2}E}{\dd q^{2}}
   - 2\frac{\dd^{2}F}{\dd p·\dd q}
    + \frac{\dd^{2}G}{\dd p^{2}}
  \right)\Add{.}
\end{multline*}


\Article{12.}

Since we always have
\[
dx^{2} + dy^{2} + dz^{2} = E\, dp^{2} + 2F\, dp·dq + G\, dq^{2},
\]
it is clear that
\[
\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}
\]
is the general expression for the linear element on the curved surface. The analysis
developed in the preceding article thus shows us that for finding the measure of curvature
there is no need of finite formulæ, which express the coordinates $x$,~$y$,~$z$ as
functions of the indeterminates $p$,~$q$; but that the general expression for the magnitude
of any linear element is sufficient. Let us proceed to some applications of this very
important theorem.

Suppose that our surface can be developed upon another surface, curved or plane,
so that to each point of the former surface, determined by the coordinates $x$,~$y$,~$z$, will
correspond a definite point of the latter surface, whose coordinates are $x'$,~$y'$,~$z'$. Evidently
$x'$,~$y'$,~$z'$ can also be regarded as functions of the indeterminates $p$,~$q$, and therefore
for the element $\Sqrt{dx'^{2} + dy'^{2} + dz'^{2}}$ we shall have an expression of the form
\[
\Sqrt{E'\, dp^{2} + 2F'\, dp·dq + G'\, dq^{2}}\Add{,}
\]
where $E'$,~$F'$,~$G'$ also denote functions of $p$,~$q$. But from the very notion of the \emph{development}
of one surface upon another it is clear that the elements corresponding to one
another on the two surfaces are necessarily equal. Therefore we shall have identically
\[
E = E',\quad F = F',\quad G = G'.
\]
Thus the formula of the preceding article leads of itself to the remarkable

\begin{Theorem}
If a curved surface is developed upon any other surface whatever, the
measure of curvature in each point remains unchanged.
\end{Theorem}
\PageSep{21}

Also it is evident that
\begin{Theorem}[]
any finite part whatever of the curved surface will retain the
same integral curvature after development upon another surface.
\end{Theorem}

Surfaces developable upon a plane constitute the particular case to which geometers
have heretofore restricted their attention. Our theory shows at once that the
measure of curvature at every point of such surfaces is equal to zero. Consequently,
if the nature of these surfaces is defined according to the third method, we shall have
at every point
\[
\frac{\dd^{2} z}{\dd x^{2}}·\frac{\dd^{2} z}{\dd y^{2}}
  - \left(\frac{\dd^{2}z}{\dd x·\dd y}\right)^{2} = 0\Add{,}
\]
a criterion which, though indeed known a short time ago, has not, at least to our
knowledge, commonly been demonstrated with as much rigor as is desirable.


\Article{13.}

What we have explained in the preceding article is connected with a particular
method of studying surfaces, a very worthy method which may be thoroughly developed
by geometers. When a surface is regarded, not as the boundary of a solid, but
as a flexible, though not extensible solid, one dimension of which is supposed to
vanish, then the properties of the surface depend in part upon the form to which we
can suppose it reduced, and in part are absolute and remain invariable, whatever may
be the form into which the surface is bent. To these latter properties, the study of
which opens to geometry a new and fertile field, belong the measure of curvature and
the integral curvature, in the sense which we have given to these expressions. To
these belong also the theory of shortest lines, and a great part of what we reserve to
be treated later. From this point of view, a plane surface and a surface developable
on a plane, \eg,~cylindrical surfaces, conical surfaces,~etc., are to be regarded as essentially
identical; and the generic method of defining in a general manner the nature of
the surfaces thus considered is always based upon the formula
\[
\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}},
\]
which connects the linear element with the two indeterminates $p$,~$q$. But before following
this study further, we must introduce the principles of the theory of shortest
lines on a given curved surface.


\Article{14.}

The nature of a curved line in space is generally given in such a way that the
coordinates $x$,~$y$,~$z$ corresponding to the different points of it are given in the form of
functions of a single variable, which we shall call~$w$. The length of such a line from
\PageSep{22}
an arbitrary initial point to the point whose coordinates are $x$,~$y$,~$z$, is expressed by
the integral
\[
%[** TN: Round outer parentheses in the original]
\int dw·\SQRT{
    \left(\frac{dx}{dw}\right)^{2}
  + \left(\frac{dy}{dw}\right)^{2}
  + \left(\frac{dz}{dw}\right)^{2}}\Add{.}
\]
If we suppose that the position of the line undergoes an infinitely small variation, so
that the coordinates of the different points receive the variations $\delta x$,~$\delta y$,~$\delta z$, the variation
of the whole length becomes
\[
\int \frac{dx·d\, \delta x + dy·d\, \delta y + dz·d\, \delta z}
          {\Sqrt{dx^{2} + dy^{2} + dz^{2}}}\Add{,}
\]
which expression we can change into the form\Note{15}
\begin{multline*}
\frac{dx·\delta x + dy·\delta y + dz·\delta z}
     {\Sqrt{dx^{2} + dy^{2} + dz^{2}}} \\
-\int \Biggl(
    \delta x·d\frac{dx}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
  + \delta y·d\frac{dy}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
  + \delta z·d\frac{dz}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
  \Biggr)\Add{.}\NoteMark
\end{multline*}
We know that, in case the line is to be the shortest between its end points, all that
stands under the integral sign must vanish. Since the line must lie on the given
surface, whose nature is defined by the equation
\[
P\, dx + Q\, dy + R\, dz = 0,
\]
the variations $\delta x$,~$\delta y$,~$\delta z$ also must satisfy the equation
\[
P\, \delta x + Q\, \delta y + R\, \delta z = 0,
\]
and from this it follows at once, according to well-known rules, that the differentials
\[
d\frac{dx}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad
d\frac{dy}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad
d\frac{dz}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
\]
must be proportional to the quantities $P$,~$Q$,~$R$ respectively. Let $dr$~be the element
of the curved line; $\lambda$~the point on the sphere representing the direction of this element;
$L$~the point on the sphere representing the direction of the normal to the curved
surface; finally, let $\xi$,~$\eta$,~$\zeta$ be the coordinates of the point~$\lambda$, and $X$,~$Y$,~$Z$ be those of
the point~$L$ with reference to the centre of the sphere. We shall then have
\[
dx = \xi\, dr,\quad
dy = \eta\, dr,\quad
dz = \zeta\, dr\Add{,}
\]
from which we see that the above differentials become $d\xi$,~$d\eta$,~$d\zeta$. And since the
quantities $P$,~$Q$,~$R$ are proportional to $X$,~$Y$,~$Z$, the character of shortest lines is
expressed by the equations
\[
\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}\Add{.}
\]
\PageSep{23}
Moreover, it is easily seen that
\[
\Sqrt{d\xi^{2} + d\eta^{2} + d\zeta^{2}}
\]
is equal to the small arc on the sphere which measures the angle between the directions
of the tangents at the beginning and at the end of the element~$dr$, and is thus
equal to~$\dfrac{dr}{\rho}$, if $\rho$~denotes the radius of curvature of the shortest line at this point.
Thus we shall have
\[
\rho\, d\xi = X\, dr,\quad
\rho\, d\eta = Y\, dr,\quad
\rho\, d\zeta = Z\, dr\Add{.}
\]


\Article{15.}

Suppose that an infinite number of shortest lines go out from a given point~$A$
on the curved surface, and suppose that we distinguish these lines from one another
by the angle that the first element of each of them makes with the first element of
one of them which we take for the first. Let $\phi$~be that angle, or, more generally, a
function of that angle, and $r$~the length of such a shortest line from the point~$A$ to
the point whose coordinates are $x$,~$y$,~$z$. Since to definite values of the variables $r$,~$\phi$
there correspond definite points of the surface, the coordinates $x$,~$y$,~$z$ can be regarded
as functions of $r$,~$\phi$. We shall retain for the notation $\lambda$, $L$, $\xi$,~$\eta$,~$\zeta$, $X$,~$Y$,~$Z$ the same
meaning as in the preceding article, this notation referring to any point whatever on
any one of the shortest lines.

All the shortest lines that are of the same length~$r$ will end on another line
whose length, measured from an arbitrary initial point, we shall denote by~$v$. Thus $v$~can
be regarded as a function of the indeterminates $r$,~$\phi$, and if $\lambda'$~denotes the point
on the sphere corresponding to the direction of the element~$dv$, and also $\xi'$,~$\eta'$,~$\zeta'$
denote the coordinates of this point with reference to the centre of the sphere, we
shall have
\[
\frac{\dd x}{\dd\phi} = \xi'·\frac{\dd v}{\dd\phi},\quad
\frac{\dd y}{\dd\phi} = \eta'·\frac{\dd v}{\dd\phi},\quad
\frac{\dd z}{\dd\phi} = \zeta'·\frac{\dd v}{\dd\phi}\Add{.}
\]
From these equations and from the equations
\[
\frac{\dd x}{\dd r} = \xi,\quad
\frac{\dd y}{\dd r} = \eta,\quad
\frac{\dd z}{\dd r} = \zeta
\]
we have
\[
\frac{\dd x}{\dd r}·\frac{\dd x}{\dd\phi} +
\frac{\dd y}{\dd r}·\frac{\dd y}{\dd\phi} +
\frac{\dd z}{\dd r}·\frac{\dd z}{\dd\phi}
  = (\xi\xi' + \eta\eta' + \zeta\zeta')·\frac{\dd v}{\dd\phi}
  = \cos \lambda\lambda'·\frac{\dd v}{\dd\phi}\Add{.}
\]
\PageSep{24}
Let $S$~denote the first member of this equation, which will also be a function of $r$,~$\phi$.
Differentiation of~$S$ with respect to~$r$ gives
\begin{align*}
\frac{\dd S}{\dd r}
  &= \frac{\dd^{2} x}{\dd r^{2}}·\frac{\dd x}{\dd\phi}
   + \frac{\dd^{2} y}{\dd r^{2}}·\frac{\dd y}{\dd\phi}
   + \frac{\dd^{2} z}{\dd r^{2}}·\frac{\dd z}{\dd\phi}
   + \tfrac{1}{2}·\frac{\dd\left(
       \biggl(\dfrac{\dd x}{\dd r}\biggr)^{2}
     + \biggl(\dfrac{\dd y}{\dd r}\biggr)^{2}
     + \biggl(\dfrac{\dd z}{\dd r}\biggr)^{2}
   \right)}{\dd \phi} \\
  &= \frac{\dd\xi}{\dd r}·\frac{\dd x}{\dd\phi}
   + \frac{\dd\eta}{\dd r}·\frac{\dd y}{\dd\phi}
   + \frac{\dd\zeta}{\dd r}·\frac{\dd z}{\dd\phi}
   + \tfrac{1}{2}·\frac{\dd(\xi^{2} + \eta^{2} + \zeta^{2})}{\dd\phi}\Add{.}
\end{align*}
But
\[
\xi^{2} + \eta^{2} + \zeta^{2} = 1,
\]
and therefore its differential is equal to zero; and by the preceding article we have,
if $\rho$~denotes the radius of curvature of the line~$r$,
\[
\frac{\dd\xi}{\dd r} = \frac{X}{\rho},\quad
\frac{\dd\eta}{\dd r} = \frac{Y}{\rho},\quad
\frac{\dd\zeta}{\dd r} = \frac{Z}{\rho}\Add{.}
\]
Thus we have
\[
\frac{\dd S}{\dd r}
  = \frac{1}{\rho}·(X\xi' + Y\eta' + Z\zeta')·\frac{\dd v}{\dd\phi}
  = \frac{1}{\rho}·\cos L\lambda'·\frac{\dd v}{\dd\phi} = 0
\]
since $\lambda'$~evidently lies on the great circle whose pole is~$L$. From this we see that
$S$~is independent of~$r$, and is, therefore, a function of $\phi$~alone. But for $r = 0$ we evidently
have $v = 0$, consequently $\dfrac{\dd v}{\dd\phi} = 0$, and $S = 0$ independently of~$\phi$. Thus, in general,
we have necessarily $S = 0$, and so $\cos\lambda\lambda' = 0$, \ie, $\lambda\lambda' = 90°$. From this follows the

\begin{Theorem}
If on a curved surface an infinite number of shortest lines of equal length
be drawn from the same initial point, the lines joining their extremities will be normal to
each of the lines.
\end{Theorem}

We have thought it worth while to deduce this theorem from the fundamental
property of shortest lines; but the truth of the theorem can be made apparent without
any calculation by means of the following reasoning. Let $AB$,~$AB'$ be two
shortest lines of the same length including at~$A$ an infinitely small angle, and let us
suppose that one of the angles made by the element~$BB'$ with the lines $BA$,~$B'A$
differs from a right angle by a finite quantity. Then, by the law of continuity, one
will be greater and the other less than a right angle. Suppose the angle at~$B$ is
equal to~$90° - \omega$, and take on the line~$AB$ a point~$C$, such that
\[
BC = BB'·\cosec \omega.
\]
Then, since the infinitely small triangle~$BB'C$ may be regarded as plane, we shall have
\[
CB' = BC·\cos \omega,
\]
\PageSep{25}
and consequently
\[
AC + CB' = AC + BC·\cos \omega
  = AB - BC·(1- \cos \omega)
  = AB' - BC·(1 - \cos \omega),
\]
\ie, the path from $A$~to~$B'$ through the point~$C$ is shorter than the shortest line,
\QEA


\Article{16.}

%[** TN: In-line theorem, marked non-semantically]
With the theorem of the preceding article we associate another, which we state
as follows: \textit{If on a curved surface we imagine any line whatever, from the different points
of which are drawn at right angles and toward the same side an infinite number of shortest
lines of the same length, the curve which joins their other extremities will cut each of the
lines at right angles.} For the demonstration of this theorem no change need be made
in the preceding analysis, except that $\phi$~must denote the length of the \emph{given} curve
measured from an arbitrary point; or rather, a function of this length. Thus all of
the reasoning will hold here also, with this modification, that $S = 0$ for $r = 0$ is
now implied in the hypothesis itself. Moreover, this theorem is more general than
the preceding one, for we can regard it as including the first one if we take for the
given line the infinitely small circle described about the centre~$A$. Finally, we may
say that here also geometric considerations may take the place of the analysis, which,
however, we shall not take the time to consider here, since they are sufficiently
obvious.


\Article{17.}

We return to the formula
\[
\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}},
\]
which expresses generally the magnitude of a linear element on the curved surface,
and investigate, first of all, the geometric meaning of the coefficients $E$,~$F$,~$G$. We
have already said in \Art{5} that two systems of lines may be supposed to lie on the
curved surface, $p$~being variable, $q$~constant along each of the lines of the one system;
and $q$~variable, $p$~constant along each of the lines of the other system. Any point
whatever on the surface can be regarded as the intersection of a line of the first
system with a line of the second; and then the element of the first line adjacent to
this point and corresponding to a variation~$dp$ will be equal to~$\sqrt{E}·dp$, and the
element of the second line corresponding to the variation~$dq$ will be equal to~$\sqrt{G}·dq$.
Finally, denoting by~$\omega$ the angle between these elements, it is easily seen that we
shall have
\[
\cos \omega = \frac{F}{\sqrt{EG}}.
\]
\PageSep{26}
Furthermore, the area of the surface element in the form of a parallelogram between
the two lines of the first system, to which correspond $q$,~$q + dq$, and the two lines of
the second system, to which correspond $p$,~$p + dp$, will be
\[
\Sqrt{EG - F^{2}}\, dp·dq.
\]

Any line whatever on the curved surface belonging to neither of the two systems
is determined when $p$~and~$q$ are supposed to be functions of a new variable, or
one of them is supposed to be a function of the other. Let $s$~be the length of such
a curve, measured from an arbitrary initial point, and in either direction chosen as
positive. Let $\theta$~denote the angle which the element
\[
ds = \Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}
\]
makes with the line of the first system drawn through the initial point of the element,
and, in order that no ambiguity may arise, let us suppose that this angle is
measured from that branch of the first line on which the values of~$p$ increase, and is
taken as positive toward that side toward which the values of~$q$ increase. These conventions
being made, it is easily seen that
\begin{align*}
\cos \theta·ds &= \sqrt{E}·dp + \sqrt{G}·\cos \omega·dq
  = \frac{E\, dp + F\, dq}{\sqrt{E}}\Add{,} \\
\sin \theta·ds &= \sqrt{G}·\sin \omega·dq
  = \frac{\sqrt{(EG - F^{2})}·dq}{\sqrt{E}}\Add{.}
\end{align*}


\Article{18.}

We shall now investigate the condition that this line be a shortest line. Since
its length~$s$ is expressed by the integral
\[
s = \int \Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}\Add{,}
\]
the condition for a minimum requires that the variation of this integral arising from
an infinitely small change in the position become equal to zero. The calculation, for
our purpose, is more simply made in this case, if we regard $p$ as a function of~$q$.
When this is done, if the variation is denoted by the characteristic~$\delta$, we have
\begin{align*}
\delta s &= \int \frac{\left(
    \dfrac{\dd E}{\dd p}·dp^{2}
  + 2\dfrac{\dd F}{\dd p}·dp·dq
  + \dfrac{\dd G}{\dd p}·dq^{2}
  \right) \delta p
  + (2E\, dp + 2F\, dq)\, d\, \delta p}{2\, ds} \displaybreak[1] \\
  &= \frac{E\, dp + F\, dq}{ds}·\delta p \\
\PageSep{27}
  &\qquad+ \int \delta p \left(\frac{
    \dfrac{\dd E}{\dd p}·dp^{2}
  + 2\dfrac{\dd F}{\dd p}·dp·dq
  + \dfrac{\dd G}{\dd p}·dq^{2}}{2\, ds}
  - d·\frac{E\, dp + F\, dq}{ds}\right)
\end{align*}
and we know that what is included under the integral sign must vanish independently
of~$\delta p$. Thus we have
\begin{multline*}
\frac{\dd E}{\dd p}·dp^{2}
  + 2\frac{\dd F}{\dd p}·dp·dq
  + \frac{\dd G}{\dd p}·dq^{2}
  = 2\, ds·d·\frac{E\, dp + F\, dq}{ds} \\
\begin{aligned}
&= 2\, ds·d·\left(\sqrt{E}·\cos\theta\right) \\ %[** TN: Added parentheses]
&= \frac{ds·dE·\cos\theta}{\sqrt{E}}
 - 2\, ds·d\theta·\sqrt{E}·\sin\theta \\
%** Translator's note from corrigenda: The original and the Latin reprints **
%** lack the factor 2; the correction is made in all the translations.     **
&= \frac{(E\, dp + F\, dq)\, dE}{E} - \Erratum{}{2}\Sqrt{EG - F^{2}}·\Erratum{dp}{dq}·d\theta \\
&= \left(\frac{E\, dp + F\, dq}{E}\right)
 ·\left(\frac{\dd E}{\dd p}·dp + \frac{\dd E}{\dd q}·dq\right)
  - 2\Sqrt{EG - F^{2}}·dq·d\theta\Add{.}
\end{aligned}
\end{multline*}
This gives the following conditional equation for a shortest line:
\begin{multline*}
\Sqrt{EG - F^{2}}·d\theta
  = \frac{1}{2}·\frac{F}{E}·\frac{\dd E}{\dd p}·dp
  + \frac{1}{2}·\frac{F}{E}·\frac{\dd E}{\dd q}·dq
  + \frac{1}{2}·\frac{\dd E}{\dd q}·dp \\
  - \frac{\dd F}{\dd p}·dp - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{,}
\end{multline*}
which can also be written
\[
\Sqrt{EG - F^{2}}·d\theta
  = \frac{1}{2}·\frac{F}{E}·dE
  + \frac{1}{2}·\frac{\dd E}{\dd q}·dp
  - \frac{\dd F}{\dd p}·dp
  - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{.}
\]
From this equation, by means of the equation
\[
\cot\theta = \frac{E}{\Sqrt{EG - F^{2}}}·\frac{dp}{dq}
  + \frac{F}{\Sqrt{EG - F^{2}}}\Add{,}
\]
it is also possible to eliminate the angle~$\theta$, and to derive a differential equation of
the second order between $p$~and~$q$, which, however, would become more complicated
and less useful for applications than the preceding.


\Article{19.}

The general formulæ, which we have derived in \Arts{11}{18} for the measure of
curvature and the variation in the direction of a shortest line, become much simpler
if the quantities $p$,~$q$ are so chosen that the lines of the first system cut everywhere
\PageSep{28}
orthogonally the lines of the second system; \ie, in such a way that we have generally
$\omega = 90°$, or $F = 0$. Then the formula for the measure of curvature becomes
\[
4E^{2}G^{2}k
  = E·\frac{\dd E}{\dd q}·\frac{\dd G}{\dd q}
  + E\left(\frac{\dd G}{\dd p}\right)^{2}
  + G·\frac{\dd E}{\dd p}·\frac{\dd G}{\dd p}
  + G\left(\frac{\dd E}{\dd q}\right)^{2}
  - 2EG\left(\frac{\dd^{2} E}{\dd q^{2}} + \frac{\dd^{2} G}{\dd p^{2}}\right),
\]
and for the variation of the angle~$\theta$
\[
\sqrt{EG}·d\theta
  = \frac{1}{2}·\frac{\dd E}{\dd q}·dp
  - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{.}
\]

Among the various cases in which we have this condition of orthogonality, the
most important is that in which all the lines of one of the two systems, \eg, the
first, are shortest lines. Here for a constant value of~$q$ the angle~$\theta$ becomes equal to
zero, and therefore the equation for the variation of~$\theta$ just given shows that we must
have $\dfrac{\dd E}{\dd q} = 0$, or that the coefficient~$E$ must be independent of~$\Erratum{g}{q}$; \ie, $E$~must be
either a constant or a function of $p$~alone. It will be simplest to take for~$p$
the length of each line of the first system, which length, when all the lines of the
first system meet in a point, is to be measured from this point, or, if there is no
common intersection, from any line whatever of the second system. Having made
these conventions, it is evident that $p$~and~$q$ denote now the same quantities that
were expressed in \Arts{15}{16} by $r$~and~$\phi$, and that $E = 1$. Thus the two preceding
formulæ become:
\begin{align*}
4G^{2}k
  &= \left(\frac{\dd G}{\dd p}\right)^{2} - 2G\, \frac{\dd^{2} G}{\dd p^{2}}\Add{,} \\
\sqrt{G}·d\theta
  &= -\frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{;}
\end{align*}
or, setting $\sqrt{G} = m$,
\[
k = -\frac{1}{m}·\frac{\dd^{2} m}{\dd p^{2}},\quad
d\theta = -\frac{\dd m}{\dd p}·dq\Add{.}
\]
Generally speaking, $m$~will be a function of $p$,~$q$, and $m\, dq$~the expression for the element
of any line whatever of the second system. But in the particular case where
all the lines~$p$ go out from the same point, evidently we must have $m = 0$ for $p = 0$.
Furthermore, in the case under discussion we will take for~$q$ the angle itself which
the first element of any line whatever of the first system makes with the element of
any one of the lines chosen arbitrarily. Then, since for an infinitely small value of~$p$
the element of a line of the second system (which can be regarded as a circle
described with radius~$p$) is equal to~$p\, dq$, we shall have for an infinitely small value
of~$p$, $m = p$, and consequently, for $p = 0$, $m = 0$ at the same time, and $\dfrac{\dd m}{\dd p} = 1$.
\PageSep{29}


\Article{20.}

We pause to investigate the case in which we suppose that $p$~denotes in a general
manner the length of the shortest line drawn from a fixed point~$A$ to any other
point whatever of the surface, and $q$~the angle that the first element of this line
makes with the first element of another given shortest line going out from~$A$. Let
$B$ be a definite point in the latter line, for which $q = 0$, and $C$~another definite point
of the surface, at which we denote the value of~$q$ simply by~$A$. Let us suppose the
points $B$,~$C$ joined by a shortest line, the parts of which, measured from~$B$, we denote
in a general way, as in \Art{18}, by~$s$; and, as in the same article, let us denote by~$\theta$
the angle which any element~$ds$ makes with the element~$dp$; finally, let us denote
by $\theta°$,~$\theta'$ the values of the angle~$\theta$ at the points $B$,~$C$. We have thus on the curved
surface a triangle formed by shortest lines. The angles of this triangle at $B$~and~$C$
we shall denote simply by the same letters, and $B$~will be equal to~$180° - \theta$, $C$~to $\theta'$~itself.
But, since it is easily seen from our analysis that all the angles are supposed
to be expressed, not in degrees, but by numbers, in such a way that the angle $57°\, 17'\, 45''$,
to which corresponds an arc equal to the radius, is taken for the unit, we must set
\[
\theta° = \pi - B,\quad \theta' = C\Add{,}
\]
where $2\pi$~denotes the circumference of the sphere. Let us now examine the integral
curvature of this triangle, which is equal to
\[
\int k\, d\sigma,
\]
$d\sigma$~denoting a surface element of the triangle. Wherefore, since this element is expressed
by~$m\, dp·dq$, we must extend the integral
\[
\iint \Typo{}{k}m\, dp·dq
\]
over the whole surface of the triangle. Let us begin by integration with respect to~$p$,
which, because
\[
k = -\frac{1}{m}·\frac{\dd^{2} m}{\dd p^{2}},
\]
gives
\[
dq·\left(\text{const.} - \frac{\dd m}{\dd p}\right),
\]
for the integral curvature of the area lying between the lines of the first system, to
which correspond the values $q$,~$q + dq$ of the second indeterminate. Since this integral
\PageSep{30}
curvature must vanish for $p = 0$, the constant introduced by integration must be
equal to the value of~$\dfrac{\dd m}{\dd q}$ for $p = 0$, \ie,~equal to unity. Thus we have
\[
dq\left(1 - \frac{\dd m}{\dd p}\right),
\]
where for $\dfrac{\dd m}{\dd p}$ must be taken the value corresponding to the end of this area on the
line~$CB$. But on this line we have, by the preceding article,
\[
\frac{\dd m}{\dd q}·dq = -d\theta,
\]
whence our expression is changed into $dq + d\theta$. Now by a second integration, taken
from $q = 0$ to $q = A$, we obtain for the integral curvature
\[
A + \theta'- \theta°,
\]
or
\[
A + B + C - \pi.
\]

The integral curvature is equal to the area of that part of the sphere which corresponds
to the triangle, taken with the positive or negative sign according as the
curved surface on which the triangle lies is concavo-concave or concavo-convex. For
unit area will be taken the square whose side is equal to unity (the radius of the
sphere), and then the whole surface of the sphere becomes equal to~$4\pi$. Thus the
part of the surface of the sphere corresponding to the triangle is to the whole surface
of the sphere as $±(A + B + C - \pi)$ is to~$4\pi$. This theorem, which, if we mistake
not, ought to be counted among the most elegant in the theory of curved surfaces,
may also be stated as follows:

\begin{Theorem}[]
The excess over~$180°$ of the sum of the angles of a triangle formed by shortest lines
on a concavo-concave curved surface, or the deficit from~$180°$ of the sum of the angles of
a triangle formed hy shortest lines on a concavo-convex curved surface, is measured by the
area of the part of the sphere which corresponds, through the directions of the normals, to
that triangle, if the whole surface of the sphere is set equal to $720$~degrees.
\end{Theorem}

More generally, in any polygon whatever of $n$~sides, each formed by a shortest
line, the excess of the sum of the angles over $(2n - 4)$~right angles, or the deficit from
$(2n - 4)$~right angles (according to the nature of the curved surface), is equal to the
area of the corresponding polygon on the sphere, if the whole surface of the sphere is
set equal to $720$~degrees. This follows at once from the preceding theorem by dividing
the polygon into triangles.
\PageSep{31}


\Article{21.}

Let us again give to the symbols $p$,~$q$, $E$,~$F$,~$G$, $\omega$ the general meanings which
were given to them above, and let us further suppose that the nature of the curved
surface is defined in a similar way by two other variables, $p'$,~$q'$, in which case the
general linear element is expressed by
\[
\Sqrt{E'\, dp'^{2} + 2F'\, dp'·dq' + G'\, dq'^{2}}\Add{.}
\]
Thus to any point whatever lying on the surface and defined by definite values of
the variables $p$,~$q$ will correspond definite values of the variables $p'$,~$q'$, which will
therefore be functions of $p$,~$q$. Let us suppose we obtain by differentiating them
\begin{alignat*}{2}
dp' &= \alpha\, dp &&+ \beta\,  dq\Add{,} \\
dq' &= \gamma\, dp &&+ \delta\, dq\Add{.}
\end{alignat*}
We shall now investigate the geometric meaning of the coefficients $\alpha$,~$\beta$,~$\gamma$,~$\delta$.

Now \emph{four} systems of lines may thus be supposed to lie upon the curved surface,
for which $p$,~$q$, $p'$,~$q'$ respectively are constants. If through the definite point to
which correspond the values $p$,~$q$, $p'$,~$q'$ of the variables we suppose the four lines
belonging to these different systems to be drawn, the elements of these lines, corresponding
to the positive increments $dp$,~$dq$, $dp'$,~$dq'$, will be
\[
\sqrt{E}·dp,\quad
\sqrt{G}·dq,\quad
\sqrt{E'}·dp',\quad
\sqrt{G'}·dq'.
\]
The angles which the directions of these elements make with an arbitrary fixed direction
we shall denote by $M$,~$N$, $M'$,~$N'$, measuring them in the sense in which the
second is placed with respect to the first, so that $\sin(N - M)$ is positive. Let us
suppose (which is permissible) that the fourth is placed in the same sense with respect
to the third, so that $\sin(N' - M')$ also is positive. Having made these conventions,
if we consider another point at an infinitely small distance from the first point, and
to which correspond the values $p + dp$, $q + dq$, $p' + dp'$, $q' + dq'$ of the variables, we
see without much difficulty that we shall have generally, \ie, independently of the
values of the increments $dp$,~$dq$, $dp'$,~$dq'$,
\[
\sqrt{E}·dp·\sin M + \sqrt{G}·dq·\sin N
 = \sqrt{E'}·dp'·\sin M' + \sqrt{G'}·dq'·\sin N'\Add{,}
\]
since each of these expressions is merely the distance of the new point from the line
from which the angles of the directions begin. But we have, by the notation introduced
above,
\[
N - M = \omega.
\]
In like manner we set
\[
N' - M' = \omega',
\]
\PageSep{32}
and also
\[
N - M' = \psi.
\]
Then the equation just found can be thrown into the following form:
\begin{multline*}
\sqrt{E}·dp · \sin(M' - \omega + \psi) + \sqrt{G}·dq·\sin(M' + \psi) \\
= \sqrt{E'}·dp'·\sin M' + \sqrt{G'}·dq'·\sin(M' + \omega')\Add{,}
\end{multline*}
or
\begin{multline*}
\sqrt{E}·dp·\sin(N' - \omega - \omega' + \psi)
  - \sqrt{G}·dq·\sin(N' - \omega' + \psi) \\
  = \sqrt{E'}·dp'·\sin(N' - \omega') + \sqrt{G'}·dq'·\sin N'\Add{.}
\end{multline*}
And since the equation evidently must be independent of the initial direction, this
direction can be chosen arbitrarily. Then, setting in the second formula $N' = 0$, or in
the first $M' = 0$, we obtain the following equations:
\begin{align*}
\sqrt{E'}·\sin \omega'·dp'
  &= \sqrt{E}·\sin(\omega + \omega' - \psi)·dp
   + \sqrt{G}·\sin(\omega' - \psi)·dq\Add{,} \\
\sqrt{G'}·\sin \omega'·dq'
  &= \sqrt{E}·\sin(\psi - \omega)·dp + \sqrt{G}·\sin\psi·dq\Add{;}
\end{align*}
and these equations, since they must be identical with
\begin{alignat*}{2}
dp' &= \alpha\, dp &&+ \beta\,  dq\Add{,} \\
dq' &= \gamma\, dp &&+ \delta\, dq\Add{,}
\end{alignat*}
determine the coefficients $\alpha$,~$\beta$,~$\gamma$,~$\delta$. We shall have
\begin{align*}
\alpha &= \sqrt{\frac{E}{E'}}·\frac{\sin(\omega + \omega' - \psi)}{\sin\omega'}, &
\beta  &= \sqrt{\frac{G}{E'}}·\frac{\sin(\omega' - \psi)}{\sin\omega'}\Add{,} \\
\gamma &= \sqrt{\frac{E}{G'}}·\frac{\sin(\psi - \omega)}{\sin\omega'}, &
\delta &= \sqrt{\frac{G}{G'}}·\frac{\sin\psi}{\sin\omega'}\Add{.}
\end{align*}
These four equations, taken in connection with the equations
\begin{align*}
\cos\omega  &= \frac{F}{\sqrt{EG}}, &
\cos\omega' &= \frac{F'}{\sqrt{E'G'}}, \\
\sin\omega  &= \sqrt{\frac{EG - F^{2}}{EG}}, &
\sin\omega' &= \sqrt{\frac{E'G' - F'^{2}}{E'G'}},
\end{align*}
may be written
\begin{align*}
\alpha\Sqrt{E'G' - F'^{2}} &= \sqrt{EG'}·\sin(\omega + \omega' - \psi)\Add{,} \\
\beta \Sqrt{E'G' - F'^{2}} &= \sqrt{GG'}·\sin(\omega' - \psi)\Add{,} \\
\gamma\Sqrt{E'G' - F'^{2}} &= \sqrt{EE'}·\sin(\psi - \omega)\Add{,} \\
\delta\Sqrt{E'G' - F'^{2}} &= \sqrt{GE'}·\sin \psi\Add{.}
\end{align*}

Since by the substitutions
\begin{alignat*}{2}
dp' &= \alpha\, dp &&+ \beta\,  dq, \\
dq' &= \gamma\, dp &&+ \delta\, dq\Add{,}
\end{alignat*}
\PageSep{33}
the trinomial
\[
E'\, dp'^{2} + 2F'\, dp'·dq' + G'\, dq'^{2}
\]
is transformed into
\[
E\, dp^{2} + 2F\, dp·dq + G\, dq^{2},
\]
we easily obtain
\[
EG - F^{2} = (E'G' - F'^{2})(\alpha\delta - \beta\gamma)^{2}\Add{;}
\]
and since, \textit{vice versa}, the latter trinomial must be transformed into the former by the
substitution
\[
(\alpha\delta - \beta\gamma)\, dp =  \delta\, dp' - \beta\, dq',\quad
(\alpha\delta - \beta\gamma)\, dq = -\gamma\, dp' + \alpha\, dq',
\]
we find\Note{16}
\begin{align*}
E\delta^{2} - 2F\gamma\delta + G\gamma^{2}
  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·E'\Add{,} \\
%
-E\beta\delta + F(\alpha\delta + \beta\gamma) - G\alpha\gamma
  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·F'\Add{,}\NoteMark \\
%
E\beta^{2} - 2F\alpha\beta + G\alpha^{2}
  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·G'\Add{.}
\end{align*}


\Article{22.}

From the general discussion of the preceding article we proceed to the very
extended application in which, while keeping for $p$,~$q$ their most general meaning, we
take for $p'$,~$q'$ the quantities denoted in \Art{15} by $r$,~$\phi$. We shall use $r$,~$\phi$ here
also in such a way that, for any point whatever on the surface, $r$~will be the shortest
distance from a fixed point, and $\phi$~the angle at this point between the first element
of~$r$ and a fixed direction. We have thus
\[
E' = 1,\quad
F' = 0,\quad
\omega' = 90°.
\]
Let us set also
\[
\sqrt{G'} = m,
\]
so that any linear element whatever becomes equal to
\[
\Sqrt{dr^{2} + m^{2}\, d\phi^{2}}.
\]
Consequently, the four equations deduced in the preceding article for $\alpha$,~$\beta$,~$\gamma$,~$\delta$ give
\begin{align*}
\sqrt{E}·\cos(\omega - \psi) = \frac{\dd r}{\dd p}\Add{,}
\Tag{(1)} \\
\sqrt{G}·\cos \psi = \frac{\dd r}{\dd q}\Add{,}
\Tag{(2)} \displaybreak[1] \\
\PageSep{34}
\sqrt{E}·\sin(\psi - \omega) = m·\frac{\dd\phi}{\dd p}\Add{,}
\Tag{(3)} \\
\sqrt{G}·\sin\psi = m·\frac{\dd\phi}{\dd q}\Add{.}
\Tag{(4)}
\end{align*}
But the last and the next to the last equations of the preceding article give
\begin{gather*}
EG - F^{2}
  = E\left(\frac{\dd r}{\dd q}\right)^{2}
  - 2F·\frac{\dd r}{\dd p}·\frac{\dd r}{\dd q}
  + G\left(\frac{\dd r}{\dd p}\right)^{2}\Add{,}
\Tag{(5)} \\
\left(E·\frac{\dd r}{\dd q} - F·\frac{\dd r}{\dd p}\right)·\frac{\dd\phi}{\dd q}
  = \left(F·\frac{\dd r}{\dd q} - G·\frac{\dd r}{\dd p}\right)·\frac{\dd\phi}{\dd p}\Add{.}
\Tag{(6)}
\end{gather*}

From these equations must be determined the quantities $r$,~$\phi$,~$\psi$ and (if need be)~$m$,
as functions of $p$~and~$q$. Indeed, integration of equation~(5) will give~$r$; $r$~being
found, integration of equation~(6) will give~$\phi$; and one or other of equations (1),~(2)
will give $\psi$~itself. Finally, $m$~is obtained from one or other of equations (3),~(4).

The general integration of equations (5),~(6) must necessarily introduce two arbitrary
functions. We shall easily understand what their meaning is, if we remember
that these equations are not limited to the case we are here considering, but are
equally valid if $r$~and~$\phi$ are taken in the more general sense of \Art{16}, so that $r$~is
the length of the shortest line drawn normal to a fixed but arbitrary line, and $\phi$~is
an arbitrary function of the length of that part of the fixed line which is intercepted
between any shortest line and an arbitrary fixed point. The general solution must
embrace all this in a general way, and the arbitrary functions must go over into
definite functions only when the arbitrary line and the arbitrary functions of its
parts, which $\phi$~must represent, are themselves defined. In our case an infinitely
small circle may be taken, having its centre at the point from which the distances~$r$
are measured, and $\phi$~will denote the parts themselves of this circle, divided by the
radius. Whence it is easily seen that the equations (5),~(6) are quite sufficient for
our case, provided that the functions which they leave undefined satisfy the condition
which $r$~and~$\phi$ satisfy for the initial point and for points at an infinitely small
distance from this point.

Moreover, in regard to the integration itself of the equations (5),~(6), we know
that it can be reduced to the integration of ordinary differential equations, which, however,
often happen to be so complicated that there is little to be gained by the reduction.
On the contrary, the development in series, which are abundantly sufficient for
practical requirements, when only a finite portion of the surface is under consideration,
presents no difficulty; and the formulæ thus derived open a fruitful source for
\PageSep{35}
the solution of many important problems. But here we shall develop only a single
example in order to show the nature of the method.


\Article{23.}

We shall now consider the case where all the lines for which $p$~is constant are
shortest lines cutting orthogonally the line for which $\phi = 0$, which line we can regard
as the axis of abscissas. Let $A$~be the point for which $r = 0$, $D$~any point whatever
on the axis of abscissas, $AD = p$, $B$~any point whatever on the shortest line normal
to~$AD$ at~$D$, and $BD = q$, so that $p$~can be regarded as the abscissa, $q$~the ordinate
of the point~$B$. The abscissas we assume positive on the branch of the axis of
abscissas to which $\phi = 0$ corresponds, while we always regard~$r$ as positive. We take
the ordinates positive in the region in which $\phi$~is measured between $0$~and~$180°$.

By the theorem of \Art{16} we shall have
\[
\omega = 90°,\quad
F = 0,\quad
G = 1,
\]
and we shall set also
\[
\sqrt{E} = n.
\]
Thus $n$~will be a function of $p$,~$q$, such that for $q = 0$ it must become equal to unity.
The application of the formula of \Art{18} to our case shows that on any shortest
line \emph{whatever} we must have\Note{17}
\[
d\theta = \frac{\dd n}{\dd q}·dp,\NoteMark
\]
where $\theta$~denotes the angle between the element of this line and the element of the
line for which $q$~is constant. Now since the axis of abscissas is itself a shortest line,
and since, for it, we have everywhere $\theta = 0$, we see that for $q = 0$ we must have
everywhere
\[
\frac{\dd n}{\dd q} = 0.
\]
Therefore we conclude that, if $n$~is developed into a series in ascending powers of~$q$,
this series must have the following form:
\[
n = 1 + fq^{2} + gq^{3} + hq^{4} + \text{etc.}\Add{,}
\]
where $f$,~$g$,~$h$,~etc., will be functions of~$p$, and we set
\begin{alignat*}{4}
f &= f° &&+ f'p &&+ f''p^{2} &&+ \text{etc.}\Add{,} \\
g &= g° &&+ g'p &&+ g''p^{2} &&+ \text{etc.}\Add{,} \\
h &= h° &&+ h'p &&+ h''p^{2} &&+ \text{etc.}\Add{,}
\end{alignat*}
\PageSep{36}
or
\begin{alignat*}{2}
n = 1 + f°q^{2} &+ f'pq^{2}   &&+ f''p^{2}q^{2} + \text{etc.} \\
                  &+ g° q^{3} &&+ g'pq^{3} + \text{etc.} \\
                  &             &&+ h°q^{4} + \text{etc.\ etc.}
\end{alignat*}


\Article{24.}

The equations of \Art{22} give, in our case,
\begin{gather*}
 n\sin\psi = \frac{\dd r}{\dd p},\quad
  \cos\psi = \frac{\dd r}{\dd q},\quad
-n\cos\psi = m·\frac{\dd\phi}{\dd r},\quad
  \sin\psi = m·\frac{\dd\phi}{\dd q}, \\
%
n^{2} = n^{2}\left(\frac{\dd r}{\dd q}\right)^{2}
          + \left(\frac{\dd r}{\dd p}\right)^{2},\quad
n^{2}·\frac{\dd r}{\dd q}·\frac{\dd\phi}{\dd q}
      + \frac{\dd r}{\dd p}·\frac{\dd\phi}{\dd p} = 0\Add{.}
\end{gather*}
By the aid of these equations, the fifth and sixth of which are contained in the others,
series can be developed for $r$,~$\phi$,~$\psi$,~$m$, or for any functions whatever of these quantities.
We are going to establish here those series that are especially worthy of
attention.

Since for infinitely small values of $p$,~$q$ we must have
\[
r^{2} = p^{2} + q^{2},
\]
the series for~$r^{2}$ will begin with the terms $p^{2} + q^{2}$. We obtain the terms of higher
order by the method of undetermined coefficients,\footnote
  {We have thought it useless to give the calculation here, which can be somewhat abridged by
  certain artifices.}
by means of the equation
\[
\left(\frac{1}{n}·\frac{\dd(r^{2})}{\dd p}\right)^{2}
  + \left(\frac{\dd(r^{2})}{\dd q}\right)^{2} = 4r^{2}\Add{.}
\]
Thus we have\Note{18}
\begin{alignat*}{3}
\Tag{[1]}
r^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q^{2} &&+ \tfrac{1}{2}f'p^{3}q^{2} &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2}\quad\text{etc.} \\
      &+ q^{2} &&+ \tfrac{1}{2}g°p^{2}q^{3} &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
      &&& &&+(\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}\Add{.}
\end{alignat*}

Then we have, from the formula\Note{19}
\[
r\sin\psi = \frac{1}{2n}·\frac{\dd(r^{2})}{\dd p},
\]
\begin{alignat*}{2}
\Tag{[2]}
r\sin\psi = p - \tfrac{1}{3}f°pq^{2}
  & -\tfrac{1}{4}f'p^{2}q^{2} &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\quad\text{etc.} \\
  & -\tfrac{1}{2}g°pq^{3} &&-\tfrac{2}{5}g'p^{2}q^{3} \\
  &&& -(\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})pq^{4}\Add{;}
\end{alignat*}
\PageSep{37}
and from the formula\Note{20}
\[
r\cos\psi = \tfrac{1}{2}\, \frac{\dd(r^{2})}{\dd q}\Add{,}
\]
\begin{alignat*}{2}
\Tag{[3]}
r\cos\psi = q + \tfrac{2}{3}f°p^{2}q
  & +\tfrac{1}{2}f'p^{3}q &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\quad\text{etc.} \\
  & +\tfrac{3}{4}g°p^{2}q^{2} &&+\tfrac{3}{5}g'p^{3}q^{2} \\
  &&& +(\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\Add{.}
\end{alignat*}
These formulæ give the angle~$\psi$. In like manner, for the calculation of the angle~$\phi$,
series for $r\cos\phi$ and $r\sin\phi$ are very elegantly developed by means of the partial
differential equations
\begin{align*}
&\frac{\dd·r\cos\phi}{\dd p}
  = n\cos\phi·\sin\psi - r\sin\phi·\frac{\dd\phi}{\dd p}\Add{,} \\
&\frac{\dd·r\cos\phi}{\dd q}
  = \Z\cos\phi·\cos\psi - r\sin\phi·\frac{\dd\phi}{\dd q}\Add{,} \\
&\frac{\dd·r\sin\phi}{\dd p}
  = n\sin\phi·\sin\psi + r\cos\phi·\frac{\dd\phi}{\dd p}\Add{,} \\
&\frac{\dd·r\sin\phi}{\dd q}
  = \Z\sin\phi·\cos\psi + r\cos\phi·\frac{\dd\phi}{\dd q}\Add{,} \\
&n\cos\psi·\frac{\dd\phi}{\dd q} + \sin\psi·\frac{\dd\phi}{\dd p} = 0\Add{.}
\end{align*}
A combination of these equations gives
\begin{alignat*}{3}
&\frac{r\sin\psi}{n}·\frac{\dd·r\cos\phi}{\dd p}
  &&+ r\cos\psi·\frac{\dd·r\cos\phi}{\dd q}
  &&= r\cos\phi\Add{,} \\
&\frac{r\sin\psi}{n}·\frac{\dd·r\sin\phi}{\dd p}
  &&+ r\cos\psi·\frac{\dd·r\sin\phi}{\dd q}
  &&= r\sin\phi\Add{.}
\end{alignat*}
From these two equations series for $r\cos\phi$, $r\sin\phi$ are easily developed, whose first
terms must evidently be $p$,~$q$ respectively. The series are\Note{21}
\begin{alignat*}{3}
\Tag{[4]}
r\cos\phi &= p + \tfrac{2}{3}f°pq^{2}
  &&+ \tfrac{5}{12}f'p^{2}q^{2}
  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\quad\text{etc.} \\
 &&&+ \tfrac{1}{2}g°pq^{3}
  &&+ \tfrac{7}{20}g'p^{2}q^{3} \\
&&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}\Add{,} \displaybreak[1] \\
\Tag{[5]}
r\sin\phi &= q - \tfrac{1}{3}f°p^{2}q
  &&- \tfrac{1}{6}f'p^{3}q
  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q\quad\text{etc.} \\
 &&&- \tfrac{1}{4}g°p^{2}q^{2}
  &&- \tfrac{3}{20}g'p^{3}q^{2} \\
&&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}\Add{.}
\end{alignat*}
From a combination of equations [2],~[3],~[4],~[5] a series for $r^{2}\cos(\psi + \phi)$, may
be derived, and from this, dividing by the series~[1], a series for $\cos(\psi + \phi)$, from
\PageSep{38}
which may be found a series for the angle $\psi + \phi$ itself. However, the same series
can be obtained more elegantly in the following manner. By differentiating the first
and second of the equations introduced at the beginning of this article, we obtain
\[
\sin\psi·\frac{\dd n}{\dd q}
  + n\cos\psi·\frac{\dd\psi}{\dd q}
  +  \sin\psi·\frac{\dd\psi}{\dd p} = 0\Add{,}
\]
and this combined with the equation
\[
n\cos\psi·\frac{\dd\phi}{\dd q} + \sin\psi·\frac{\dd\phi}{\dd p} = 0
\]
gives
\[
\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q}
  + \frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
  + r\cos\psi·\frac{\dd(\psi + \phi)}{\dd q} = 0\Add{.}
\]
From this equation, by aid of the method of undetermined coefficients, we can easily
derive the series for $\psi + \phi$, if we observe that its first term must be~$\frac{1}{2}\pi$, the radius
being taken equal to unity and $2\pi$~denoting the circumference of the circle,\Note{22}
\begin{alignat*}{2}
\Tag{[6]}
\psi + \phi = \tfrac{1}{2}\pi - f°pq
  &- \tfrac{2}{3}f'p^{2}q
  &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q\quad\text{etc.} \\
  &- g°pq^{2}
  &&- \tfrac{3}{4}g'p^{2}q^{2} \\
  &&&- (h° - \tfrac{1}{3}{f°}^{2})pq^{3}\Add{.}
\end{alignat*}

It seems worth while also to develop the area of the triangle~$ABD$ into a series.
For this development we may use the following conditional equation, which is easily
derived from sufficiently obvious geometric considerations, and in which $S$~denotes the
required area:\Note{23}
\[
\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p}
  + r\cos\psi·\frac{\dd S}{\dd q}
  = \frac{r\sin\psi}{n}·\int n\, dq\Add{,}\NoteMark
\]
the integration beginning with $q = 0$. From this equation we obtain, by the method
of undetermined coefficients,\Note{24}
\begin{alignat*}{3}
\Tag{[7]}
S = \tfrac{1}{2}pq
  &- \tfrac{1}{12}f°p^{3}q
  &&- \tfrac{1}{20}f'p^{4}q
  &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q\quad\text{etc.} \\
  &- \tfrac{1}{12}f°pq^{3}
  &&- \tfrac{3}{40}g°p^{3}q^{2}
  &&- \tfrac{1}{20}g'p^{4}q^{2} \\
  &&&- \tfrac{7}{120}f'p^{2}q^{3}
  &&- (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
  &&&- \tfrac{1}{10}g°pq^{4}
  &&- \tfrac{3}{40}g'p^{2}q^{4} \\
  &&&&&- (\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}\Add{.}
\end{alignat*}
\PageSep{39}


\Article{25.}

From the formulæ of the preceding article, which refer to a right triangle formed
by shortest lines, we proceed to the general case. Let $C$ be another point on the
same shortest line~$DB$, for which point~$p$ remains the same as for the point~$B$, and
$q'$,~$r'$, $\phi'$,~$\psi'$, $S'$ have the same meanings as $q$,~$r$, $\phi$,~$\psi$, $S$ have for the point~$B$. There
will thus be a triangle between the points $A$,~$B$,~$C$, whose angles we denote by
$A$,~$B$,~$C$, the sides opposite these angles by $a$,~$b$,~$c$, and the area by~$\sigma$. We represent
the measure of curvature at the points $A$,~$B$,~$C$ by $\alpha$,~$\beta$,~$\gamma$ respectively. And then
supposing (which is permissible) that the quantities $p$,~$q$,~$q - q'$ are positive, we shall
have
\begin{align*}
A &= \phi - \phi', & B &= \psi, & C &= \pi - \psi', && \\
a &= q - q',       & b &= r',   & c &= r,           & \sigma &= S - S'.
\end{align*}

We shall first express the area~$\sigma$ by a series. By changing in~[7] each of the
quantities that refer to~$B$ into those that refer to~$C$, we obtain a formula for~$S'$.
Whence we have, exact to quantities of the sixth order,\Note{25}
\begin{align*}
\sigma = \tfrac{1}{2}p(q - q')
  \bigl(1 &- \tfrac{1}{6} f°(p^{2} + q^{2} + qq' + q'^{2}) \\
          &- \tfrac{1}{60}f'p(6p^{2} + 7q^{2} + 7qq' + 7q'^{2}) \\
          &- \tfrac{1}{20}g°(q + q')(3p^{2} + 4q^{2} + 4q'^{2})\bigr)\Add{.}\NoteMark
\end{align*}
This formula, by aid of series~[2], namely,
\[
c\sin B = p(1 - \tfrac{1}{3}f°q^{2}
              - \tfrac{1}{4}f'pq^{2}
              - \tfrac{1}{2}g°q^{3} - \text{etc.})
\]
can be changed into the following:
\begin{align*}
\sigma = \tfrac{1}{2}ac\sin B
  \bigl(1 &- \tfrac{1}{6} f°(p^{2} - q^{2} + qq' + q'^{2}) \\
          &- \tfrac{1}{60}f'p(6p^{2} - 8q^{2} + 7qq' + 7q'^{2}) \\
          &- \tfrac{1}{20}g°
    (3p^{2}q + 3p^{2}q' - 6p^{3} + 4q^{2}q' + 4qq'^{2} + 4q'^{3})\bigr)\Add{.}
\end{align*}

The measure of curvature for any point whatever of the surface becomes (by \Art{19},
where $m$,~$p$,~$q$ were what $n$,~$q$,~$p$ are here)
\begin{align*}
k &= -\frac{1}{n}·\frac{\dd^{2} n}{\dd q^{2}}
   = -\frac{2f + 6gq + 12hq^{2} + \text{etc.}}{1 + fq^{2} + \text{etc.}} \\
  &= -2f - 6gq - (12h - 2f^{2}) q^{2} - \text{etc.}
\end{align*}
Therefore we have, when $p$,~$q$ refer to the point~$B$,
\[
\beta = - 2f° - 2f'p - 6g°q - 2f''p^{2} - 6g'pq - (12h° - 2{f°}^{2})q^{2} - \text{etc.}
\]
\PageSep{40}
Also
\begin{align*}
\gamma &= -2f° - 2f'p - 6g°q' - 2f''p^{2} - 6g'pq' - (12h° - 2{f°}^{2})q'^{2} - \text{etc.}\Add{,} \\
\alpha &= -2f°\Add{.}
\end{align*}
Introducing these measures of curvature into the expression for~$\sigma$, we obtain the following
expression, exact to quantities of the sixth order (exclusive):\Note{26}
\begin{align*}
\sigma = \tfrac{1}{2}ac \sin B
  \bigl(1 &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} + 3qq' + 3q'^{2}) \\
          &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} + 6qq' + 3q'^{2}) \\
          &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} + \Z qq' + 4q'^{2})\bigr)\Add{.}\NoteMark
\end{align*}
The same precision will remain, if for $p$,~$q$,~$q'$ we substitute $c\sin B$, $c\cos B$, $c\cos B - a$.
This gives\Note{27}
\begin{align*}
\Tag{[8]}
\sigma = \tfrac{1}{2}ac\sin B
  \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \Z9ac\cos B) \\
          &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\
          &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \Z9ac\cos B)\bigr)\Add{.}
\end{align*}
Since all expressions which refer to the line~$AD$ drawn normal to~$BC$ have disappeared
from this equation, we may permute among themselves the points $A$,~$B$,~$C$ and
the expressions that refer to them. Therefore we shall have, with the same precision,
\begin{align*}
\Tag{[9]}
\sigma = \tfrac{1}{2}bc\sin A
  \bigl(1 &+ \tfrac{1}{120}\alpha(3b^{2} + 3c^{2} -  12bc\cos A) \\
          &+ \tfrac{1}{120}\beta (3b^{2} + 4c^{2} - \Z9bc\cos A) \\
          &+ \tfrac{1}{120}\gamma(4b^{2} + 3c^{2} - \Z9bc\cos A)\bigr)\Add{,} \\
%
\Tag{[10]}
\sigma = \tfrac{1}{2}ab\sin C
  \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4b^{2} - \Z9ab\cos C) \\
          &+ \tfrac{1}{120}\beta (4a^{2} + 3b^{2} - \Z9ab\cos C) \\
          &+ \tfrac{1}{120}\gamma(3a^{2} + 3b^{2} -  12ab\cos C)\bigr)\Add{.}
\end{align*}


\Article{26.}

The consideration of the rectilinear triangle whose sides are equal to $a$,~$b$,~$c$ is of
great advantage. The angles of this triangle, which we shall denote by $A^{*}$,~$B^{*}$,~$C^{*}$,
differ from the angles of the triangle on the curved surface, namely, from $A$,~$B$,~$C$,
by quantities of the second order; and it will be worth while to develop these differences
accurately. However, it will be sufficient to show the first steps in these more
tedious than difficult calculations.

Replacing in formulæ [1],~[4],~[5] the quantities that refer to~$B$ by those that
refer to~$C$, we get formulæ for $r'^{2}$,~$r'\cos\phi'$, $r'\sin\phi'$. Then the development of the
expression
\PageSep{41}
\begin{align*}
  r^{2} + r'^{2} &- (q - q')^{2}
    - 2r\cos\phi·r'\cos\phi'
    - 2r\sin\phi·r'\sin\phi' \\
    &\quad= b^{2} + c^{2} - a^{2} - 2bc\cos A \\
    &\quad= 2bc(\cos A^{*} - \cos A),
\end{align*}
combined with the development of the expression
\[
r\sin\phi·r'\cos\phi' - r\cos\phi·r'\sin\phi' = bc\sin A,
\]
gives the following formula:
\begin{align*}
\cos A^{*} - \cos A
  = -(q - q')p\sin A
  \bigl(\tfrac{1}{3}f° &+ \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') \\
  &+ (\tfrac{1}{10}f'' - \tfrac{1}{45}{f°}^{2})p^{2}
   + \tfrac{3}{20}g'p(q + q') \\
  &+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})(q^{2} + qq' + q'^{2})
   + \text{etc.}\bigr)
\end{align*}
From this we have, to quantities of the fifth order,\Note{28}
\begin{align*}
A^{*} - A = +(q - q')p
  \bigl(\tfrac{1}{3}f°
  &+ \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
  &+ \tfrac{3}{20}g'p(q + q') + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
  &- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 12qq' + 7q'^{2})\bigr)\Add{.}\NoteMark
\end{align*}
Combining this formula with
\[
2\sigma = ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})- \text{etc.}\bigr)
\]
and with the values of the quantities $\alpha$,~$\beta$,~$\gamma$ found in the preceding article, we obtain,
to quantities of the fifth order,\Note{29}
\begin{align*}
\Tag{[11]}
A^{*} = A - \sigma\bigl(\tfrac{1}{6}\alpha
  &+ \tfrac{1}{12}\beta
   + \tfrac{1}{12}\gamma + \tfrac{2}{15}f''p^{2} + \tfrac{1}{5}g'p(q + q') \\
  &+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2}) \\
  &+ \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr)\Add{.}
\end{align*}
By precisely similar operations we derive\Note{30}
\begin{align*}
\Tag{[12]}
B^{*} = B - \sigma\bigl(\tfrac{1}{12}\alpha
  &+ \tfrac{1}{6}\beta
   + \tfrac{1}{12}\gamma + \tfrac{1}{10}f''p^{2} + \tfrac{1}{10}g'p(2q + q') \\
  &+ \tfrac{1}{5}h°(4q^{2} - 4qq' + 3q'^{2}) \\
  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} - 8qq' + 11q'^{2})\bigr)\Add{,}
\displaybreak[1] \\
%
\Tag{[13]}
C^{*} = C - \sigma\bigl(\tfrac{1}{12}\alpha
  &+ \tfrac{1}{12}\beta
   + \tfrac{1}{6}\gamma + \tfrac{1}{10}f''p^{2} + \tfrac{1}{10}g'p(q + 2q') \\
  &+ \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr)\Add{.}
\end{align*}
From these formulæ we deduce, since the sum $A^{*} + B^{*} + C^{*}$ is equal to two right
angles, the excess of the sum~$A + B + C$ over two right angles, namely,\Note{31}
\begin{align*}
\Tag{[14]}
A + B + C= \pi + \sigma\bigl(\tfrac{1}{3}\alpha
  &+ \tfrac{1}{3}\beta
   + \tfrac{1}{3}\gamma + \tfrac{1}{3}f''p^{2} + \tfrac{1}{2}g'p(q + q') \\
  &+ (2h° - \tfrac{1}{3}{f°}^{2})(q^{2} - qq' + q'^{2})\bigr)\Add{.}
\end{align*}
This last equation could also have been derived from formula~[6].
\PageSep{42}


\Article{27.}

If the curved surface is a sphere of radius~$R$, we shall have
\[
\alpha = \beta = \gamma = -2f° = \frac{1}{R^{2}};\quad
f'' = 0,\quad
g' = 0,\quad
6h° - {f°}^{2} = 0,
\]
or
\[
h° = \frac{1}{24R^{4}}.
\]
Consequently, formula~[14] becomes
\[
A + B + C = \pi + \frac{\sigma}{R^{2}},
\]
which is absolutely exact. But formulæ [11],~[12],~[13] give
\begin{align*}
A^{*} &= A - \frac{\sigma}{3R^{2}}
  - \frac{\sigma}{180R^{4}}(2p^{2} - q^{2} + 4qq' - q'^{2})\Add{,} \\
B^{*} &= B - \frac{\sigma}{3R^{2}}
  + \frac{\sigma}{180R^{4}}(p^{2} - 2q^{2} + 2qq' + q'^{2})\Add{,} \\
C^{*} &= C - \frac{\sigma}{3R^{2}}
  + \frac{\sigma}{180R^{4}}(p^{2} + q^{2} + 2qq' - 2q'^{2})\Add{;}
\end{align*}
or, with equal exactness,
\begin{align*}
A^{*} &= A - \frac{\sigma}{3R^{2}}
  - \frac{\sigma}{180R^{4}}(b^{2} + c^{2} - 2a^{2})\Add{,} \\
B^{*} &= B - \frac{\sigma}{3R^{2}}
  - \frac{\sigma}{180R^{4}}(a^{2} + c^{2} - 2b^{2})\Add{,} \\
C^{*} &= C - \frac{\sigma}{3R^{2}}
  - \frac{\sigma}{180R^{4}}(a^{2} + b^{2} - 2c^{2})\Add{.}
\end{align*}
Neglecting quantities of the fourth order, we obtain from the above the well-known
theorem first established by the illustrious Legendre.


\Article{28.}

Our general formulæ, if we neglect terms of the fourth order, become extremely
simple, namely:
\begin{align*}
A^{*} &= A - \tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\Add{,} \\
B^{*} &= B - \tfrac{1}{12}\sigma(\alpha + 2\beta + \gamma)\Add{,} \\
C^{*} &= C - \tfrac{1}{12}\sigma(\alpha + \beta + 2\gamma)\Add{.}
\end{align*}
\PageSep{43}

Thus to the angles $A$,~$B$,~$C$ on a non-spherical surface, unequal reductions must
be applied, so that the sines of the changed angles become proportional to the sides
opposite. The inequality, generally speaking, will be of the third order; but if the
surface differs little from a sphere, the inequality will be of a higher order. Even in
the greatest triangles on the earth's surface, whose angles it is possible to measure,
the difference can always be regarded as insensible. Thus, \eg, in the greatest of
the triangles which we have measured in recent years, namely, that between the
points Hohehagen, Brocken, Inselberg, where the excess of the sum of the angles was
$14''. 85348$, the calculation gave the following reductions to be applied to the angles:
\setlength{\TmpLen}{2in}%
\begin{align*}
\Dotrow{Hohehagen}{-4''.95113\rlap{\Add{,}}} \\
\Dotrow{Brocken}{- 4''.95104\rlap{\Add{,}}} \\
\Dotrow{Inselberg}{-4''.95131\rlap{.}}
\end{align*}


\Article{29.}

We shall conclude this study by comparing the area of a triangle on a curved
surface with the area of the rectilinear triangle whose sides are $a$,~$b$,~$c$. We shall
denote the area of the latter by~$\sigma^{*}$; hence
\[
\sigma^{*} = \tfrac{1}{2}bc\sin A^{*}
           = \tfrac{1}{2}ac\sin B^{*}
           = \tfrac{1}{2}ab\sin C^{*}\Add{.}
\]

We have, to quantities of the fourth order,
\[
\sin A^{*} = \sin A - \tfrac{1}{12}\sigma\cos A·(2\alpha + \beta + \gamma)\Add{,}
\]
or, with equal exactness,
\[
\sin A = \sin A^{*}·\bigl(1 + \tfrac{1}{24}bc\cos A·(2\alpha + \beta + \gamma)\bigr)\Add{.}
\]
Substituting this value in formula~[9], we shall have, to quantities of the sixth order,
\begin{align*}
\sigma = \tfrac{1}{2}bc\sin A^{*}·\bigl(1
  &+ \tfrac{1}{120}\alpha(3b^{2}+ 3c^{2} - 2bc\cos A) \\
  &+ \tfrac{1}{120}\beta (3b^{2}+ 4c^{2} - 4bc\cos A) \\
  &+ \tfrac{1}{120}\gamma(4b^{2}+ 3c^{2} - 4bc\cos A)\bigr),
\end{align*}
or, with equal exactness,
\[
\sigma = \sigma^{*}\bigl(1
  + \tfrac{1}{120}\alpha(a^{2} + 2b^{2} + 2c^{2})
  + \tfrac{1}{120}\beta (2a^{2} + b^{2} + 2c^{2})
  + \tfrac{1}{120}\gamma(2a^{2} + 2b^{2} + c^{2})\Add{.}
\]
For the sphere this formula goes over into the following form:
\[
\sigma = \sigma^{*}\bigl(1 + \tfrac{1}{24}\alpha(a^{2} + b^{2} + c^{2})\bigr).
\]
\PageSep{44}
It is easily verified that, with the same precision, the following formula may be taken
instead of the above:
\[
\sigma
  = \sigma^{*}\sqrt{\frac{\sin A·\sin B·\sin C}
                         {\sin A^{*}·\sin B^{*}·\sin C^{*}}}\Add{.}
\]
If this formula is applied to triangles on non-spherical curved surfaces, the error, generally
speaking, will be of the fifth order, but will be insensible in all triangles such
as may be measured on the earth's surface.
\PageSep{45}


\Abstract{\small GAUSS'S ABSTRACT OF THE DISQUISITIONES GENERALES CIRCA \\
SUPERFICIES CURVAS, PRESENTED TO THE ROYAL \\
SOCIETY OF GÖTTINGEN. \\
\tb \\[8pt]

\footnotesize\textsc{Göttingische gelehrte Anzeigen.} No.~177. Pages 1761--1768. 1827. November~5.
}

On the 8th~of October, Hofrath Gauss presented to the Royal Society a paper:
\begin{center}
\textit{Disquisitiones generales circa superficies curvas.}
\end{center}

Although geometers have given much attention to general investigations of curved
surfaces and their results cover a significant portion of the domain of higher geometry,
this subject is still so far from being exhausted, that it can well be said that, up to
this time, but a small portion of an exceedingly fruitful field has been cultivated.
Through the solution of the problem, to find all representations of a given surface upon
another in which the smallest elements remain unchanged, the author sought some
years ago to give a new phase to this study. The purpose of the present discussion
is further to open up other new points of view and to develop some of the new truths
which thus become accessible. We shall here give an account of those things which
can be made intelligible in a few words. But we wish to remark at the outset that
the new theorems as well as the presentations of new ideas, if the greatest generality
is to be attained, are still partly in need of some limitations or closer determinations,
which must be omitted here.

In researches in which an infinity of directions of straight lines in space is concerned,
it is advantageous to represent these directions by means of those points upon
a fixed sphere, which are the end points of the radii drawn parallel to the lines. The
centre and the radius of this \emph{auxiliary sphere} are here quite arbitrary. The radius may
be taken equal to unity. This procedure agrees fundamentally with that which is constantly
employed in astronomy, where all directions are referred to a fictitious celestial
sphere of infinite radius. Spherical trigonometry and certain other theorems, to which
the author has added a new one of frequent application, then serve for the solution of
the problems which the comparison of the various directions involved can present.
\PageSep{46}

If we represent the direction of the normal at each point of the curved surface by
the corresponding point of the sphere, determined as above indicated, namely, in this
way, to every point on the surface, let a point on the sphere correspond; then, generally
speaking, to every line on the curved surface will correspond a line on the sphere,
and to every part of the former surface will correspond a part of the latter. The less
this part differs from a plane, the smaller will be the corresponding part on the sphere.
It is, therefore, a very natural idea to use as the measure of the total curvature,
which is to be assigned to a part of the curved surface, the area of the corresponding
part of the sphere. For this reason the author calls this area the \emph{integral curvature} of
the corresponding part of the curved surface. Besides the magnitude of the part, there
is also at the same time its \emph{position} to be considered. And this position may be in
the two parts similar or inverse, quite independently of the relation of their magnitudes.
The two cases can be distinguished by the positive or negative sign of the
total curvature. This distinction has, however, a definite meaning only when the
figures are regarded as upon definite sides of the two surfaces. The author regards
the figure in the case of the sphere on the outside, and in the case of the curved surface
on that side upon which we consider the normals erected. It follows then that
the positive sign is taken in the case of convexo-convex or concavo-concave surfaces
(which are not essentially different), and the negative in the case of concavo-convex
surfaces. If the part of the curved surface in question consists of parts of these different
sorts, still closer definition is necessary, which must be omitted here.

The comparison of the areas of two corresponding parts of the curved surface and of
the sphere leads now (in the same manner as, \eg, from the comparison of volume and
mass springs the idea of density) to a new idea. The author designates as \emph{measure of
curvature} at a point of the curved surface the value of the fraction whose denominator is
the area of the infinitely small part of the curved surface at this point and whose numerator
is the area of the corresponding part of the surface of the auxiliary sphere, or the
integral curvature of that element. It is clear that, according to the idea of the author,
integral curvature and measure of curvature in the case of curved surfaces are analogous
to what, in the case of curved lines, are called respectively amplitude and curvature
simply. He hesitates to apply to curved surfaces the latter expressions, which
have been accepted more from custom than on account of fitness. Moreover, less
depends upon the choice of words than upon this, that their introduction shall be justified
by pregnant theorems.

The solution of the problem, to find the measure of curvature at any point of a curved
surface, appears in different forms according to the manner in which the nature of the
curved surface is given. When the points in space, in general, are distinguished by
\PageSep{47}
three rectangular coordinates, the simplest method is to express one coordinate as a function
of the other two. In this way we obtain the simplest expression for the measure of
curvature. But, at the same time, there arises a remarkable relation between this
measure of curvature and the curvatures of the curves formed by the intersections of
the curved surface with planes normal to it. \textsc{Euler}, as is well known, first showed
that two of these cutting planes which intersect each other at right angles have this
property, that in one is found the greatest and in the other the smallest radius of curvature;
or, more correctly, that in them the two extreme curvatures are found. It will
follow then from the above mentioned expression for the measure of curvature that this
will be equal to a fraction whose numerator is unity and whose denominator is the product
of the extreme radii of curvature. The expression for the measure of curvature will be
less simple, if the nature of the curved surface is determined by an equation in $x$,~$y$,~$z$.
And it will become still more complex, if the nature of the curved surface is given so that
$x$,~$y$,~$z$ are expressed in the form of functions of two new variables $p$,~$q$. In this last case
the expression involves fifteen elements, namely, the partial differential coefficients of the
first and second orders of $x$,~$y$,~$z$ with respect to $p$~and~$q$. But it is less important in itself
than for the reason that it facilitates the transition to another expression, which must be
classed with the most remarkable theorems of this study. If the nature of the curved
surface be expressed by this method, the general expression for any linear element upon
it, or for $\Sqrt{dx^{2} + dy^{2} + dz^{2}}$, has the form $\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}$, where $E$,~$F$,~$G$
are again functions of $p$~and~$q$. The new expression for the measure of curvature mentioned
above contains merely these magnitudes and their partial differential coefficients
of the first and second order. Therefore we notice that, in order to determine the
measure of curvature, it is necessary to know only the general expression for a linear
element; the expressions for the coordinates $x$,~$y$,~$z$ are not required. A direct result
from this is the remarkable theorem: If a curved surface, or a part of it, can be developed
upon another surface, the measure of curvature at every point remains unchanged
after the development. In particular, it follows from this further: Upon a curved
surface that can be developed upon a plane, the measure of curvature is everywhere
equal to zero. From this we derive at once the characteristic equation of surfaces
developable upon a plane, namely,
\[
\frac{\dd^{2} z}{\dd x^{2}}·\frac{\dd^{2} z}{\dd y^{2}}
  - \left(\frac{\dd^{2} z}{\dd x·\dd y}\right)^{2} = 0,
\]
when $z$~is regarded as a function of $x$~and~$y$. This equation has been known for some
time, but according to the author's judgment it has not been established previously
with the necessary rigor.
\PageSep{48}

These theorems lead to the consideration of the theory of curved surfaces from a
new point of view, where a wide and still wholly uncultivated field is open to investigation.
If we consider surfaces not as boundaries of bodies, but as bodies of which
one dimension vanishes, and if at the same time we conceive them as flexible but not
extensible, we see that two essentially different relations must be distinguished, namely,
on the one hand, those that presuppose a definite form of the surface in space; on the
other hand, those that are independent of the various forms which the surface may
assume. This discussion is concerned with the latter. In accordance with what has
been said, the measure of curvature belongs to this case. But it is easily seen that
the consideration of figures constructed upon the surface, their angles, their areas and
their integral curvatures, the joining of the points by means of shortest lines, and the
like, also belong to this case. All such investigations must start from this, that the
very nature of the curved surface is given by means of the expression of any linear
element in the form $\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}$. The author has embodied in the
present treatise a portion of his investigations in this field, made several years ago,
while he limits himself to such as are not too remote for an introduction, and may, to
some extent, be generally helpful in many further investigations. In our abstract, we
must limit ourselves still more, and be content with citing only a few of them as
types. The following theorems may serve for this purpose.

If upon a curved surface a system of infinitely many shortest lines of equal lengths
be drawn from one initial point, then will the line going through the end points of
these shortest lines cut each of them at right angles. If at every point of an arbitrary
line on a curved surface shortest lines of equal lengths be drawn at right angles to this
line, then will all these shortest lines be perpendicular also to the line which joins their
other end points. Both these theorems, of which the latter can be regarded as a generalization
of the former, will be demonstrated both analytically and by simple geometrical
considerations. \begin{Theorem}[]The excess of the sum of the angles of a triangle formed by shortest lines
over two right angles is equal to the total curvature of the triangle.\end{Theorem} It will be assumed here
that that angle ($57°\, 17'\, 45''$) to which an arc equal to the radius of the sphere corresponds
will be taken as the unit for the angles, and that for the unit of total curvature will be
taken a part of the spherical surface, the area of which is a square whose side is equal to
the radius of the sphere. Evidently we can express this important theorem thus also:
the excess over two right angles of the angles of a triangle formed by shortest lines is to
eight right angles as the part of the surface of the auxiliary sphere, which corresponds
to it as its integral curvature, is to the whole surface of the sphere. In general, the
excess over $2n - 4$~right angles of the angles of a polygon of $n$~sides, if these are
shortest lines, will be equal to the integral curvature of the polygon.
\PageSep{49}

The general investigations developed in this treatise will, in the conclusion, be applied
to the theory of triangles of shortest lines, of which we shall introduce only a couple of
important theorems. If $a$,~$b$,~$c$ be the sides of such a triangle (they will be regarded as
magnitudes of the first order); $A$,~$B$,~$C$ the angles opposite; $\alpha$,~$\beta$,~$\gamma$ the measures of
curvature at the angular points; $\sigma$~the area of the triangle, then, to magnitudes of the
fourth order, $\frac{1}{3}(\alpha + \beta + \gamma)\sigma$ is the excess of the sum $A + B + C$ over two right angles.
Further, with the same degree of exactness, the angles of a plane rectilinear triangle
whose sides are $a$,~$b$,~$c$, are respectively
\begin{align*}
A &- \tfrac{1}{12}(2\alpha + \beta + \gamma)\sigma\Add{,} \\
B &- \tfrac{1}{12}(\alpha + 2\beta + \gamma)\sigma\Add{,} \\
C &- \tfrac{1}{12}(\alpha + \beta + 2\gamma)\sigma.
\end{align*}
We see immediately that this last theorem is a generalization of the familiar theorem first
established by \textsc{Legendre}. By means of this theorem we obtain the angles of a plane
triangle, correct to magnitudes of the fourth order, if we diminish each angle of the corresponding
spherical triangle by one-third of the spherical excess. In the case of non-spherical
surfaces, we must apply unequal reductions to the angles, and this inequality,
generally speaking, is a magnitude of the third order. However, even if the whole surface
differs only a little from the spherical form, it will still involve also a factor denoting
the degree of the deviation from the spherical form. It is unquestionably important for
the higher geodesy that we be able to calculate the inequalities of those reductions and
thereby obtain the thorough conviction that, for all measurable triangles on the surface
of the earth, they are to be regarded as quite insensible. So it is, for example, in the
case of the greatest triangle of the triangulation carried out by the author. The greatest
side of this triangle is almost fifteen geographical\footnote
  {This German geographical mile is four minutes of arc at the equator, namely, $7.42$~kilometers,
  and is equal to about $4.6$~English statute miles. [Translators.]}
miles, and the excess of the sum
of its three angles over two right angles amounts almost to fifteen seconds. The three
reductions of the angles of the plane triangle are $4''.95113$, $4''.95104$, $4''.95131$. Besides,
the author also developed the missing terms of the fourth order in the above expressions.
Those for the sphere possess a very simple form. However, in the case of
measurable triangles upon the earth's surface, they are quite insensible. And in the
example here introduced they would have diminished the first reduction by only two
units in the fifth decimal place and increased the third by the same amount.
\PageSep{50}
%[** Blank page]
\PageSep{51}


\Notes.

%[** TN: Line numbers have been omitted]
\LineRef{1}{Art.~1, p.~3, l.~3}. Gauss got the idea of using the auxiliary sphere from astronomy.
\Cf.~Gauss's Abstract, \Pgref[p.]{abstract}.

\LineRef[1]{2}{Art.~2, p.~3, l.~2~fr.~bot}. In the Latin text \textit{situs} is used for the direction or
orientation of a plane, the position of a plane, the direction of a line, and the position
of a point.

\LineRef[2]{2}{Art.~2, p.~4, l.~14}. In the Latin texts the notation
\[
\cos(1)L^{2} + \cos(2)L^{2} + \cos(3) L^{2} = 1
\]
is used. This is replaced in the translations (except Böklen's) by the more recent
notation
\[
\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1.
\]

\LineRef[3]{2}{Art.~2, p.~4, l.~3~fr.~bot}. This stands in the original and in Liouville s reprint,
\[
\cos A (\cos t\sin t' - \sin t\cos t')(\cos t''\sin t''' - \sin t''\sin t''').
\]

%[** TN: One-off macro; hyperlink is hard-coded into macro definition]
\LineRefs{Art.~2, pp.~4--6}. Theorem~VI is original with Gauss, as is also the method of
deriving~VII\@. The following figures show the points and lines of Theorems VI~and~VII:
%[Illustrations]
\Figure{051}

\LineRef{3}{Art.~3, p.~6}. The geometric condition here stated, that the curvature be continuous
for each point of the surface, or part of the surface, considered is equivalent to
the analytic condition that the first and second derivatives of the function or functions
defining the surface be finite and continuous for all points of the surface, or
part of the surface, considered.

\LineRef[6]{4}{Art.~4, p.~7, l.~20}. In the Latin texts the notation~$XX$ for~$X^{2}$,~etc., is used.
\PageSep{52}

\LineRef[7]{4}{Art.~4, p.~7}. ``The second method of representing a surface (the expression of
the coordinates by means of two auxiliary variables) was first used by Gauss for
arbitrary surfaces in the case of the problem of conformal mapping. [Astronomische
Abhandlungen, edited by H.~C. Schumacher, vol.~III, Altona,~1825; Gauss, \Title{Werke},
vol.~IV, p.~189; reprinted in vol.~55 of Ostwald's Klassiker.---\Cf.~also Gauss, \Title{Theoria
attractionis corporum sphaer.\ ellipt.}, Comment.\ Gött.~II, 1813; Gauss, \Title{Werke}, vol.~V,
p.~10.] Here he applies this representation for the first time to the determination of
the direction of the surface normal, and later also to the study of curvature and of
geodetic lines. The geometrical significance of the variables $p$,~$q$ is discussed more fully
in \Art{17}. This method of representation forms the source of many new theorems,
of which these are particularly worthy of mention: the corollary, that the measure of
%[** TN: On next two lines, replaced "Art." by "Arts."]
curvature remains unchanged by the bending of the surface (\Arts{11}{12}); the theorems
of \Arts{15}{16} concerning geodetic lines; the theorem of \Art{20}; and, finally, the
results derived in the conclusion, which refer a geodetic triangle to the rectilinear triangle
whose sides are of the same length.'' [Wangerin.]

\LineRef{5}{Art.~5, p.~8}. ``To decide the question, which of the two systems of values found
in \Art{4} for $X$,~$Y$,~$Z$ belong to the normal directed outwards, which to the normal
directed inwards, we need only to apply the theorem of \Art{2}~(VII), provided we use
the second method of representing the surface. If, on the contrary, the surface is
defined by the equation between the coordinates $W = 0$, then the following simpler\Typo{ con-}{}
considerations lead to the answer. We draw the line~$d\sigma$ from the point~$A$ towards
the outer side, then, if $dx$,~$dy$,~$dz$ are the projections of~$d\sigma$, we have
\[
P\, dx + Q\, dy + R\, dz > 0.
\]
On the other hand, if the angle between $\sigma$~and the normal taken outward is acute,
then
\[
\frac{dx}{d\sigma}X + \frac{dy}{d\sigma}Y + \frac{dz}{d\sigma}Z > 0.
\]
This condition, since $d\sigma$~is positive, must be combined with the preceding, if the first
solution is taken for $X$,~$Y$,~$Z$. This result is obtained in a similar way, if the surface
is analytically defined by the third method.'' [Wangerin.]

\LineRef[8]{6}{Art.~6, p.~10, l.~4}. The definition of measure of curvature here given is the one
generally used. But Sophie Germain defined as a measure of curvature at a point of
a surface the sum of the reciprocals of the principal radii of curvature at that point,
or double the so-called mean curvature. \Cf.~Crelle's Journ.\ für Math., vol.~VII\@.
Casorati defined as a measure of curvature one-half the sum of the squares of the
reciprocals of the principal radii of curvature at a point of the surface. \Cf.~Rend.\
del R.~Istituto Lombardo, ser.~2, vol.~22, 1889; Acta Mathem.\ vol.~XIV, p.~95, 1890.
\PageSep{53}

\LineRef[9]{6}{Art.~6, p.~11, l.~21}. Gauss did not carry out his intention of studying the most
general cases of figures mapped on the sphere.

\LineRef{7}{Art.~7, p.~11, l.~31}. ``That the consideration of a surface element which has the
form of a triangle can be used in the calculation of the measure of curvature, follows
from this fact that, according to the formula developed on \Pageref{12}, $k$~is independent
of the magnitudes $dx$,~$dy$, $\delta x$,~$\delta y$, and that, consequently, $k$~has the same value for
every infinitely small triangle at the same point of the surface, therefore also for surface
elements of any form whatever lying at that point.'' [Wangerin.]

\LineRef[10]{7}{Art.~7, p.~12, l.~20}. The notation in the Latin text for the partial derivatives:
\[
\frac{dX}{dx},\quad \frac{dX}{dy},\quad \text{etc.},
\]
has been replaced throughout by the more recent notation:
\[
\frac{\dd X}{\dd x},\quad \frac{\dd X}{\dd y},\quad \text{etc.}
\]

\LineRef{7}{Art.~7, p.~13, l.~16}. This formula, as it stands in the original and in Liouville's
reprint, is
\[
dY = -Z^{3}tu\, dt - Z^{3}(1 + t^{2})\, du.
\]
The incorrect sign in the second member has been corrected in the reprint in Gauss,
\Title{Werke}, vol.~IV, and in the translations.

\LineRef[12]{8}{Art.~8, p.~15, l.~3}. Euler's work here referred to is found in Mem.\ de~l'Acad.\
de~Berlin, vol.~XVI, 1760.

\LineRef[13]{10}{Art.~10, p.~18, ll.~8,~9,~10}. Instead of $D$,~$D'$,~$D''$ as here defined, the Italian
geometers have introduced magnitudes denoted by the same letters and equal, in
Gauss's notation, to
\[
\frac{D}{\Sqrt{EG - F^{2}}},\quad
\frac{D'}{\Sqrt{EG - F^{2}}},\quad
\frac{D''}{\Sqrt{EG - F^{2}}}
\]
respectively.

\LineRef[14]{11}{Art.~11, p.~19, ll.~4,~6,~fr.~bot}. In the original and in Liouville's reprint, two of
these formulæ are incorrectly given:
\[
\frac{\dd F}{\dd q} = m'' + n,\quad
n = \frac{\dd F}{\dd q} - \frac{1}{2}·\frac{\dd E}{\dd q}.
\]
The proper corrections have been made in Gauss, \Title{Werke}, vol.~IV, and in the translations.

\LineRef{13}{Art.~13, p.~21, l.~20}. Gauss published nothing further on the properties of developable
surfaces.
\PageSep{54}

\LineRef[15]{14}{Art.~14, p.~22, l.~8}. The transformation is easily made by means of integration
by parts.

\LineRef{17}{Art.~17, p.~25}. If we go from the point $p$,~$q$ to the point $(p + dp, q)$, and if the
Cartesian coordinates of the first point are $x$,~$y$,~$z$, and of the second $x + dx$, $y + dy$,
$z + dz$; with $ds$~the linear element between the two points, then the direction cosines
of~$ds$ are
\[
\cos \alpha = \frac{dx}{ds},\quad
\cos \beta  = \frac{dy}{ds},\quad
\cos \gamma = \frac{dz}{ds}.
\]
Since we assume here $q = \text{Constant}$ or $dq = 0$, we have also
\[
dx = \frac{\dd x}{\dd p}·dp,\quad
dy = \frac{\dd y}{\dd p}·dp,\quad
dz = \frac{\dd z}{\dd p}·dp,\quad
ds = ±\sqrt{E}·dp.
\]
If $dp$~is positive, the change~$ds$ will be taken in the positive direction. Therefore
$ds = \sqrt{E}·dp$,
\[
\cos\alpha = \frac{1}{\sqrt{E}}·\frac{\dd x}{\dd p},\quad
\cos\beta  = \frac{1}{\sqrt{E}}·\frac{\dd y}{\dd p},\quad
\cos\gamma = \frac{1}{\sqrt{E}}·\frac{\dd z}{\dd p}\Typo{,}{.}
\]
In like manner, along the line $p = \text{Constant}$, if $\cos \alpha'$, $\cos \beta'$, $\cos \gamma'$ are the direction
cosines, we obtain
\[
\cos\alpha' = \frac{1}{\sqrt{G}}·\frac{\dd x}{\dd q},\quad
\cos\beta'  = \frac{1}{\sqrt{G}}·\frac{\dd y}{\dd q},\quad
\cos\gamma' = \frac{1}{\sqrt{G}}·\frac{\dd z}{\dd q}.
\]
And since
\begin{align*}
\cos\omega
  &= \cos\alpha \cos\alpha'
   + \cos\beta  \cos\beta'
   + \cos\gamma \cos\gamma', \\
\cos\omega
  &= \frac{F}{\sqrt{EG}}.
\end{align*}
From this follows
\[
\sin\omega = \frac{\Sqrt{EG - F^{2}}}{\sqrt{EG}}.
\]
And the area of the quadrilateral formed by the lines $p$,~$p + dp$, $q$,~$q + dq$ is
\[
d\sigma = \Sqrt{EG - F^{2}}·dp·dq.
\]

\LineRef[16]{21}{Art.~21, p.~33, l.~12}. In the original, in Liouville's reprint, in the two French
translations, and in Böklen's translation, the next to the last formula of this article
is written
\[
E\beta\delta - F(\alpha\delta + \beta\gamma) + G\alpha\gamma
  = \frac{EG - F\Typo{'}{}^{2}}{E'G' - F'^{2}}·F'\Add{.}
\]
\PageSep{55}
The proper correction in sign has been made in Gauss, \Title{Werke}, vol.~IV, and in Wangerin's
translation.

\LineRef[17]{23}{Art.~23, p.~35, l.~13~fr.~bot}. In the Latin texts and in Roger's and Böklen's
translations this formula has a minus sign on the right hand side. The correction in
sign has been made in Abadie's and Wangerin's translations.

\LineRef{23}{Art.~23, p.~35}. The figure below represents the lines and angles mentioned in
this and the following articles\Chg{:}{.}
%[Illustration]
\Figure{055}

\LineRef[18]{24}{Art.~24, p.~36}. Derivation of formula~[1].

Let
\[
r^{2} = p^{2} + q^{2} + R_{3} + R_{4} + R_{5} + R_{6} + \text{etc.}
\]
where $R_{3}$~is the aggregate of all the terms of the third degree in $p$~and~$q$, $R_{4}$~of all
the terms of the fourth degree,~etc. Then by differentiating, squaring, and omitting
terms above the sixth degree, we obtain
\begin{align*}
\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} = 4p^{2}
  &+ \left(\frac{\dd R_{3}}{\dd p}\right)^{2}
   + \left(\frac{\dd R_{4}}{\dd p}\right)^{2}
   + 4p\frac{\dd R_{3}}{\dd p} %[** TN: Omitted parentheses]
   + 4p\frac{\dd R_{4}}{\dd p} \\
  &+ 4p\frac{\dd R_{5}}{\dd p}
   + 4p\frac{\dd R_{6}}{\dd p}
   + 2p\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
   + 2p\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p},
\intertext{and}
\left(\frac{\dd(r^{2})}{\dd \Erratum{p}{q}}\right)^{2} = 4q^{2}
  &+ \left(\frac{\dd R_{3}}{\dd q}\right)^{2}
   + \left(\frac{\dd R_{4}}{\dd q}\right)^{2}
   + 4q\frac{\dd R_{3}}{\dd q}
   + 4q\frac{\dd R_{4}}{\dd q} \\
  &+ 4q\frac{\dd R_{5}}{\dd q}
   + 4q\frac{\dd R_{6}}{\dd q}
   + 2q\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}
   + 2q\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}.
\end{align*}
\PageSep{56}

Hence we have
{\small
\begin{align*}
&\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} +
\left(\frac{\dd(r^{2})}{\dd q}\right)^{2} - 4r^{2} \\
  &= 4\left(p\frac{\dd R_{3}}{\dd p} + q\frac{\dd R_{3}}{\dd q} - R_{3}\right)
   + 4\left(p\frac{\dd R_{4}}{\dd p} + q\frac{\dd R_{4}}{\dd q} - R_{4}
     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}\right) \\
  &\quad+ 4\left(p\frac{\dd R_{5}}{\dd p} + q\frac{\dd R_{5}}{\dd q} - R_{5}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}\right) \\
  &\quad+ 4\left(p\frac{\dd R_{6}}{\dd p} + q\frac{\dd R_{6}}{\dd q} - R_{6}
     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}\right)
\displaybreak[1] \\
  &= 8R_{3} + 4\left(3R_{4}
     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}\right)
   + 4\left(4R_{5}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}\right) \\
  &\quad+ 4\left(5R_{6}
     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}\right),
\end{align*}}%
since, according to a familiar theorem for homogeneous functions,
\[
p\frac{\dd R_{3}}{\dd p} + q\frac{\dd R_{3}}{\dd q} = 3R_{3},\quad\text{etc.}
\]
By dividing unity by the square of the value of~$n$, given at the end of \Art{23}, and
omitting terms above the fourth degree, we have
\[
1 - \frac{1}{n^{2}}
  = 2f°q^{2} + 2f'pq^{2} + 2g°q^{3} - 3{f°}^{2}q^{4}
    + 2f''p^{2}q^{2} + 2g'pq^{3} + 2h°q^{4}.
\]
This, multiplied by the last equation but one of the preceding page, on rejecting terms
above the sixth degree, becomes
\begin{multline*}
%[** TN: Re-broken]
\left(1 - \frac{1}{n^{2}}\right)
\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} \\
\begin{alignedat}{3}
  = 8f°p^{2}q^{2} &+ 8f'p^{3}q^{2} &&- 12{f°}^{2}p^{2}q^{4} &&+ 8h°p^{2}q^{4} \\
                 &+ 8g°p^{2}q^{3} &&+ 8f''p^{4}q^{2}
  &&+2 f°q^{2}\left(\frac{\dd R_{3}}{\dd p}\right)^{2} \\
                 &+ 8f°pq^{2} \frac{\dd R_{3}}{\dd p} &&+ 8g'p^{3}q^{3}
  &&+8 f'p^{2}q^{2} \frac{\dd R_{3}}{\dd p} + 8g°pq^{3} \frac{\dd R_{3}}{\dd p}\\
  &&&&&+ 8f°pq^{2} \frac{\dd R_{4}}{\dd p}.
\end{alignedat}
\end{multline*}
Therefore, since from the fifth equation of \Art{24}:
\[
\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} +
\left(\frac{\dd(r^{2})}{\dd q}\right)^{2} - 4r^{2}
  = \left(1 - \frac{1}{n^{2}}\right)
    \left(\frac{\dd(r^{2})}{\dd p}\right)^{2},
\]
\PageSep{57}
we have
{\small
\begin{multline*}
8R_{3} + 4\left(3R_{4}
  + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
  + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}
\right)
+ 4\left(4R_{5}
  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}
\right) \\
  + 4\left(5R_{6}
  + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
  + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}
\right) \\
\begin{aligned}
  &= 8f°p^{2}q^{2} + 8f'p^{3}q^{2} + 8g°p^{2}q^{3} + 8f°pq^{2}\frac{\dd R_{3}}{\dd p} - 12{f°}^{2}p^{2}q^{4} + 8f''p^{4}q^{2} \\
  &\quad+ 8g'p^{3}q^{3} + 8h°p^{2}q^{4} + 2f°q^{2}\left(\frac{\dd R_{3}}{\dd p}\right)^{2} + 8f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p} + 8f°pq^{2}\frac{\dd R_{4}}{\dd p} + 8g°pq^{3}\frac{\dd R_{3}}{\dd p}.
\end{aligned}
\end{multline*}}%
Whence, by the method of undetermined coefficients, we find
\begin{align*}
R_{3} &= 0,\quad
R_{4} = \tfrac{2}{3}f°p^{2}q^{2},\quad
R_{5} = \tfrac{1}{2}f'p^{3}q^{2} + \tfrac{1}{2}g°p^{2}q^{3}, \\
R_{6} &= (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2}
  + \tfrac{2}{5}g'p^{3}q^{3} + (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}.
\end{align*}
And therefore we have
\begin{alignat*}{3}
\Tag{[1]}
r^{2} &= p^{2} + \tfrac{2}{5}f°p^{2}q^{2}
  &&+ \tfrac{1}{2}f'p^{3}q^{2}
  &&+ (\tfrac{2}{5}f''- \tfrac{4}{45}{f°}^{2})p^{4}q^{2} + \text{etc.} \\
%
  &+q^{2}
  &&+ \tfrac{1}{2}g°p^{2}q^{3}
  &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}.
\end{alignat*}

This method for deriving formula~[1] is taken from Wangerin.

\LineRef[19]{24}{Art.~24, p.~36}. Derivation of formula~[2].

By taking one-half the reciprocal of the series for~$n$ given in \LineRef{23}{Art.~23, p.~36}, we
obtain
\[
\frac{1}{2n} = \tfrac{1}{2} \bigl[
  1 - f°q^{2} - f'pq^{2} - g°q^{3}
  - f''p^{2}q^{2} - g'pq^{3} - (h° - {f°}^{2})q^{4} - \text{etc.}
\bigr].
\]
And by differentiating formula~[1] with respect to~$p$, we obtain
\begin{alignat*}{2}
\frac{\dd(r^{2})}{\dd p} = 2 \bigl[p + \tfrac{1}{2}f°pq^{2}
  &+ \tfrac{3}{4}f'p^{2}q^{2}
  &&+ 2 (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{3}q^{2} \\
  &+ \tfrac{1}{2}g°pq^{3}
  &&+ \tfrac{3}{5}g'p^{2}q^{3} \\
  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4} + \text{etc.}
\bigr].
\end{alignat*}
Therefore, since
\[
r\sin\psi = \frac{1}{2n}·\frac{\dd(r^{2})}{\dd p},
\]
we have, by multiplying together the two series above,
\begin{alignat*}{2}
\Tag{[2]}
r\sin\psi = p - \tfrac{1}{3}f°pq^{2}
  &- \tfrac{1}{4}f'p^{2}q^{2}
  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2} - \text{etc.} \\
  &- \tfrac{1}{2}g°pq^{3}
  &&- \tfrac{2}{5}g'p^{2}q^{3} \\
  &&&- (\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})pq^{4}.
\end{alignat*}
\PageSep{58}

\LineRef[20]{24}{Art.~24, p.~37}. Derivation of formula~[3].

By differentiating~[1] on \Pageref{57} with respect to~$q$, we find
\begin{alignat*}{2}
\frac{\dd(r^{2})}{\dd q} = 2 \bigl[q + \tfrac{2}{3}f°p^{2}q
  &+ \tfrac{1}{2}f'p^{3}q
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
  &+ \tfrac{3}{4}g°p^{2}q^{2}
  &&+ \tfrac{3}{5}g'p^{3}q^{2}
    + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3} + \text{etc.}
\bigr].
\end{alignat*}
Therefore we have, since
\begin{gather*}
r\cos \psi = \tfrac{1}{2}\frac{\dd(r^{2})}{\dd q}, \\
\Tag{[3]}
\begin{alignedat}[t]{2}
r\cos\psi = q + \tfrac{2}{3}f°p^{2}q
  &+ \tfrac{1}{2}f'p^{3}q
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q + \text{etc.} \\
  &+ \tfrac{3}{4}g°p^{2}q^{2}
  &&+ \tfrac{3}{5}g'p^{3}q^{2}
    + (\tfrac{4}{15}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}.
\end{alignedat}
\end{gather*}

\LineRef[21]{24}{Art.~24, p.~37}. Derivation of formula~[4].

Since $r\cos\phi$ becomes equal to~$p$ for infinitely small values of $p$~and~$q$, the series
for~$r\cos\phi$ must begin with~$p$. Hence we set
\[
\Tag{(1)}
r\cos\phi = p + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.}
\]
Then, by differentiating, we obtain
\begin{alignat*}{2}
\Tag{(2)}
\frac{\dd(r\cos\phi)}{\dd p}
  &= 1 + {}&&\frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
       + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p} + \text{etc.}\Add{,} \\
\Tag{(3)}
\frac{\dd(r\cos\phi)}{\dd q}
  &= &&\frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
   + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q} + \text{etc.} \\
\end{alignat*}
By dividing~[2] \Pageref[p.]{57} by~$n$ on \Pageref{36}, we obtain
\begin{align*}
\Tag{(4)}
\frac{r\sin\psi}{n} = p - \tfrac{4}{3}f°pq^{2}
  &- \tfrac{5}{4}f'p^{2}q^{2}
   - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2} - \text{etc.} \\
  &- \tfrac{3}{2}g°pq^{3} - \tfrac{7}{5}g'p^{2}q^{3}
   - (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}.
\end{align*}
Multiplying (2) by~(4), we have
\begin{multline*}
%[** TN: Re-broken]
\Tag{(5)}
\frac{r\sin\psi}{n}·\frac{\dd(r\cos\phi)}{\dd p} \\
\begin{aligned}
  = p &+ p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
   + p\frac{\dd R_{4}}{\dd p} + p\frac{\dd R_{5}}{\dd p}
   - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q^{2} \\
  &\begin{alignedat}{3}
    - \tfrac{4}{3}f°pq^{2}
    &- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
    &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
    &&- \tfrac{7}{5}g'p^{2}q^{3} \\
    &- \tfrac{5}{4}f'p^{2}q^{2}
    &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
    &&- (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4} \\
    &\Typo{}{-}\tfrac{3}{2}g°pq^{3}
    &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
    &&- \text{etc.}
  \end{alignedat}
\end{aligned}
\end{multline*}
\PageSep{59}
Multiplying (3) by~[3] \Pageref[p.]{58}, we have
\begin{multline*}
%[** TN: Re-broken]
\Tag{(6)}
r\cos\psi·\frac{\dd(r\cos\phi)}{\dd q} \\
\begin{alignedat}{3}
  = q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
  & + q\frac{\dd R_{4}}{\dd q}
  &&+ q\frac{\dd R_{5}}{\dd q}
  &&+ \tfrac{1}{2}f'p^{3}q \frac{\dd R_{2}}{\dd q} \\
  & + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} + \text{etc.}
\end{alignedat}
\end{multline*}
Since
\[
\frac{r\sin\psi}{n}·\frac{\dd(r\cos\phi)}{\dd p}
  + r\cos\psi·\frac{\dd(r\cos\phi)}{\dd q} = r\cos\phi,
\]
we have, by setting (1)~equal to the sum of (5)~and~(6),
\begin{multline*}
p + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.} \\
\begin{alignedat}{5}
= p &+ p\frac{\dd R_{2}}{\dd p}
  &&+ p\frac{\dd R_{3}}{\dd p}
  &&+ p\frac{\dd R_{4}}{\dd p}
  &&+ p\frac{\dd R_{5}}{\dd p}
  &&- (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4} \\
%
  &+ q\frac{\dd R_{2}}{\dd q}
  &&- \tfrac{4}{3}f°pq^{2}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
  &&+ q\frac{\dd R_{5}}{\dd q} \\
%
  &&&+ q\frac{\dd R_{3}}{\dd q}
  &&- \tfrac{5}{4}f'p^{2}q^{2}
  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q} \\
%
  &&&&&- \tfrac{3}{2}g°pq^{3}
  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
  &&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q} \\
%
  &&&&&+ q\frac{\dd R_{4}}{\dd q}
  &&- (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}
  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} \\
%
  &&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
  &&- \tfrac{7}{5}g'p^{2}q^{3} + \text{etc.},
\end{alignedat}
\end{multline*}
from which we find
\begin{align*}
R_{2} &= 0,\quad
R_{3} = \tfrac{2}{3}f°pq^{2},\quad
R_{4} = \tfrac{5}{12}f'p^{2}q^{2} + \tfrac{1}{2}g°pq^{3}, \\
R_{5} &= \tfrac{7}{20}g'p^{2}q^{3}
  + (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}
  + (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}.
\end{align*}
Therefore we have finally
\begin{alignat*}{2}
\Tag{[4]}
r\cos\phi = p + \tfrac{2}{3}f°pq^{2}
  &+ \tfrac{5}{12}f'p^{2}q^{2}
  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2} + \text{etc.} \\
  &+ \tfrac{1}{2}g°pq^{3}
  &&+ \tfrac{7}{20}g'p^{2}q^{3} \\
  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}.
\end{alignat*}

\LineRef[21]{24}{Art.~24, p.~37}. Derivation of formula~[5].

Again, since $r\sin\phi$ becomes equal to~$q$ for infinitely small values of $p$~and~$q$,
we set
\[
\Tag{(1)}
r\sin\phi = q + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.}
\]
\PageSep{60}
Then we have by differentiation
\begin{alignat*}{2}
\Tag{(2)}
\frac{\dd(r\sin\phi)}{\dd p}
  &= &&\frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
   + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p} + \text{etc.} \\
\Tag{(3)}
\frac{\dd(r\sin\phi)}{\dd q}
  &= 1 + {} &&\frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
       + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q} + \text{etc.} \\
\end{alignat*}
Multiplying (4) \Pageref[p.]{58} by this~(2), we obtain
\begin{multline*}
%[** TN: Re-broken]
\Tag{(4)}
\frac{r\sin\psi}{n}·\frac{\dd(r\sin\phi)}{\dd p} \\
\begin{alignedat}{3}
  = p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
  &+ p\frac{\dd R_{4}}{\dd p}
  &&+ p\frac{\dd R_{5}}{\dd p}
  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p} \\
  & - \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p} - \text{etc.}
\end{alignedat}
\end{multline*}
Likewise from (3)~and~[3] \Pageref[p.]{58}, we obtain
\begin{multline*}
%[** TN: Re-broken]
\Tag{(5)}
r\cos\psi·\frac{\dd(r\sin\phi)}{\dd q} \\
\begin{aligned}
  = q &+ q\frac{\dd R_{2}}{\dd q}
  + q\frac{\dd R_{3}}{\dd q}
  + q\frac{\dd R_{4}}{\dd q}
  + q\frac{\dd R_{5}}{\dd q}
  + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
%
  &\begin{aligned}
   + \tfrac{2}{3}f°p^{2}q
   + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
  &+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
   - \tfrac{3}{5}g'p^{3}q^{2} \\
%
   + \tfrac{1}{2}f'p^{3}q
  &+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
   + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3} \\
%
   + \tfrac{3}{4}g°p^{2}q^{2}
  &+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} + \text{etc.}
\end{aligned}
\end{aligned}
\end{multline*}
Since
\[
\frac{r\sin\psi}{n}·\frac{\dd(r\sin\phi)}{\dd p}
  + r\cos\psi·\frac{\dd(r\sin\phi)}{\dd q} = r\sin\phi,
\]
by setting (1) equal to the sum of (4)~and~(5), we have
{\small
\begin{multline*}
q + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.} \\
\begin{alignedat}{4}
= q &+ p\frac{\dd R_{2}}{\dd p}
   &&+ p\frac{\dd R_{3}}{\dd p}
   &&+ p\frac{\dd R_{4}}{\dd p} + \tfrac{1}{2}f'p^{3}q
   &&+ p\frac{\dd R_{5}}{\dd p}
     + \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
     + q\frac{\dd R_{5}}{\dd q} + \text{etc.} \\
%
   & + q\frac{\dd R_{2}}{\dd q}
   &&+ q\frac{\dd R_{3}}{\dd q}
   &&+ q\frac{\dd R_{4}}{\dd q} + \tfrac{3}{4}g°p^{2}q^{2}
   &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
     + \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q}
     + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q} \\
%
   &&&+ \tfrac{2}{3}f°p^{2}q
   &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
   &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
     + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
%
 &&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
   &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
     + \tfrac{3}{5}g'p^{3}q^{2}
     + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\Typo{.}{,}
\end{alignedat}
\end{multline*}}%
\PageSep{61}
from which we find
\begin{align*}
R_{2} &= 0,\quad
R_{3} = -\tfrac{1}{3}f°p^{2}q,\quad
R_{4} = -\tfrac{1}{6}f'p^{3}q - \tfrac{1}{4}g°p^{2}q^{2}, \\
R_{5} &= -(\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q
  - \tfrac{3}{20}g'p^{3}q^{2}
  - (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}.
\end{align*}
Therefore, substituting these values in~(1), we have
\begin{alignat*}{2}
\Tag{[5]}
r\sin\phi = q - \tfrac{1}{3}f°p^{2}q
  &-  \tfrac{1}{6}f'p^{3}q
  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q - \text{etc.} \\
  &-  \tfrac{1}{4}g°p^{2}q^{2}
  &&- \tfrac{3}{20}g'p^{3}q^{2} \\
  &&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}.
\end{alignat*}

\LineRef[22]{24}{Art.~24, p.~38}. Derivation of formula~[6].

Differentiating $n$ on \Pageref{36} with respect to~$q$, we obtain
\begin{alignat*}{3}
\Tag{(1)}
\frac{\dd n}{\dd q} = 2f°q
  &+  2f'pq &&+ 2f''p^{2}q &&+ \text{etc.} \\
  &\Typo{}{+} 3g°q^{2} &&+ 3g'pq^{2} &&+ \text{etc.} \\
  &&&+ 4h°q^{3} &&+ \text{etc.\Typo{,}{} etc.}\Add{,}
\end{alignat*}
and hence, multiplying this series by~(4) on \Pageref{58}, we find
\begin{align*}
\Tag{(2)}
\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q} = 2f°pq
  &+ 2f'p^{2}q + 2f''p^{3}q + 3g'p^{2}q^{2} + \text{etc.} \\
  &+ 3g°pq^{2} + (4h° - \tfrac{8}{3}{f°}^{2})pq^{3}.
\end{align*}

For infinitely small values of $r$, $\psi + \phi = \dfrac{\pi}{2}$, as is evident from the figure on \Pgref{fig:055}.
Hence we set
\[
\psi + \phi = \frac{\pi}{2} + R_{1} + R_{2} + R_{3} + R_{4} + \text{etc.}
\]
Then we shall have, by differentiation,
\begin{align*}
\Tag{(3)}
\frac{\dd(\psi + \phi)}{\dd p}
  &= \frac{R_{1}}{\dd p} + \frac{R_{2}}{\dd p}
   + \frac{R_{3}}{\dd p} + \frac{R_{4}}{\dd p} + \text{etc.}\Add{,} \\
\Tag{(4)}
\frac{\dd(\psi + \phi)}{\dd q}
  &= \frac{R_{1}}{\dd q} + \frac{R_{2}}{\dd q}
   + \frac{R_{3}}{\dd q} + \frac{R_{4}}{\dd q} + \text{etc.} \\
\end{align*}
Therefore, multiplying (4) on \Pageref{58} by~(3), we find
\begin{alignat*}{2}
\Tag{(5)}
\frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
  = p\frac{R_{1}}{\dd p} + p\frac{R_{2}}{\dd p}
  &+  p\frac{R_{3}}{\dd p}
  &&+ p\frac{R_{4}}{\dd p} + \text{etc.} \\
%
  &-  \tfrac{4}{3}f°pq^{2}\frac{\dd R_{1}}{\dd p}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p} \\
%
&&&-\tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{1}}{\dd p} \\
&&&-\tfrac{3}{2}g°pq^{3}\frac{\dd R_{1}}{\dd p},
\end{alignat*}
\PageSep{62}
and, multiplying~[3] on \Pageref{58} by~(4), we find
\begin{alignat*}{2}
\Tag{(6)}
r\cos\psi·\frac{\dd(\psi + \phi)}{\dd q}
  = q\frac{R_{1}}{\dd q} + q\frac{R_{2}}{\dd q}
  &+  q\frac{R_{3}}{\dd q}
  &&+ q\frac{R_{4}}{\dd q} + \text{etc.} \\
%
  &+  \tfrac{2}{3}f°p^{2}q\frac{\dd R_{1}}{\dd q}
  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q} \\
%
&&&+\tfrac{1}{2}f'p^{3}q\frac{\dd R_{1}}{\dd q} \\
&&&+\tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{1}}{\dd q}.
\end{alignat*}
And since
\[
\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q}
  + \frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
  + r\cos\psi·\frac{\dd(\psi + \phi)}{\dd p} = 0,
\]
we shall have, by adding (2),~(5), and~(6),
\begin{alignat*}{5}
0 &= p\frac{\dd R_{1}}{\dd p}
  &&+ 2f°pq
  &&+ 2f'p^{2}q
  &&+ 2f''p^{3}q
  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{1}}{\dd p} \\
%
  &\Typo{}{+} q\frac{\dd R_{1}}{\dd q}
  &&+ p\frac{\dd R_{2}}{\dd p}
  &&+ 3g°pq^{2}
  &&+ 3g'p^{2}q^{2}
  &&+ q\frac{\dd R_{4}}{\dd q} \\
%
 &&&+ q\frac{\dd R_{2}}{\dd q}
  &&+ p\frac{\dd R_{3}}{\dd p}
  &&+ (4h° - \tfrac{8}{3}{f°})pq^{3}
  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q} \\
%
&&&&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{1}}{\dd p}
  &&+ p\frac{\dd R_{4}}{\dd p}
  && +\tfrac{1}{2}f'p^{3}q\frac{\dd R_{1}}{\dd q} \\
%
&&&&&+ q\frac{\dd R_{3}}{\dd q}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{1}}{\dd q} \\
%
&&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{1}}{\dd q}
  &&- \tfrac{5}{4}f°p^{2}q^{2}\frac{\dd R_{1}}{\dd p}
  &&+ \text{etc.}
\end{alignat*}
From this equation we find
\begin{align*}
R_{1} &= 0,\quad
R_{2} = -f°pq,\quad
R_{3} = -\tfrac{2}{3}f' p^{2}q - g°pq^{2}, \\
R_{4} &= -(\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q
  - \tfrac{3}{4}g'p^{2}q^{2} - (h° - \tfrac{1}{3}{f°}^{2})pq^{3}.
\end{align*}
Therefore we have finally
\begin{alignat*}{2}
\Tag{[6]}
\psi + \phi = \frac{\pi}{2} - f°pq
  &-  \tfrac{2}{3}f'p^{2}q
  &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q - \text{etc.} \\
  &- g°pq^{2}
  && -\tfrac{3}{4}g'p^{2}q^{2} \\
  &&&- (h° - \tfrac{1}{3}{f°}^{2})pq^{3}.
\end{alignat*}
\PageSep{63}
%XXXX

\LineRef[23]{24}{Art.~24, p.~38, l.~19}. The differential equation from which formula~[7] follows
is derived in the following manner. In the figure on \Pgref{fig:055}, prolong $AD$ to~$D'$,
making $DD' = dp$\Chg{, through}{. Through}~$D'$ perpendicular to~$AD'$ draw a geodesic line, which will
cut~$AB$ in~$B'$. Finally, take $D'B'' = DB$, so that $BB''$~is perpendicular to~$B'D'$.
Then, if by~$ABD$ we mean the area of the triangle~$ABD$,
\[
\frac{\dd S}{\dd r} = \lim \frac{AB'D' - ABD}{BB'}
  = \lim \frac{BDD'B'}{BB'}
  = \lim \frac{BDD'B''}{DD'}·\lim \frac{DD'}{BB'},
\]
since the surface $BDD'B''$ differs from $BDD'B'$ only by an infinitesimal of the
second order. And since
\[
BDD'B'' = dp·\int n\, dq,\quad\text{or}\quad
\lim \frac{BDD'B''}{DD'} = \int n\, dq,
\]
and since, further,
\[
\lim \frac{DD'}{BB'} = \frac{\dd p}{\dd r},
\]
consequently
\[
\frac{\dd S}{\dd r} = \frac{\dd p}{\dd r}·\int n\, dq.
\]
Therefore also
\[
\frac{\dd S}{\dd p}·\frac{\dd p}{\dd r} +
\frac{\dd S}{\dd q}·\frac{\dd q}{\dd r} = \frac{\dd p}{\dd r}·\int n\, dq.
\]
Finally, from the values for $\dfrac{\dd r}{\dd p}$, $\dfrac{\dd r}{\dd q}$ given at the beginning of \LineRef{24}{Art.~24, p.~36}, we have
\[
\frac{\dd p}{\dd r} = \frac{1}{n}\sin\psi,\quad
\frac{\dd q}{\dd r} = \cos\psi,
\]
so that we have
\[
\frac{\dd S}{\dd p}·\frac{\sin\psi}{n} +
\frac{\dd S}{\dd q}·\cos\psi = \frac{\sin\psi}{n}·\int n\, dq.
\]
\null\hfill[Wangerin.]

\LineRef[24]{24}{Art.~24, p.~38}. Derivation of formula~[7].

For infinitely small values of $p$~and~$q$, the area of the triangle~$ABC$ becomes
equal to $\frac{1}{2}pq$. The series for this area, which is denoted by~$S$, must therefore begin
with~$\frac{1}{2}pq$, or~$R_{2}$. Hence we put
\[
S = R_{2} + R_{3} + R_{4} + R_{5} + R_{6} + \text{etc.}
\]
\PageSep{64}
By differentiating, we obtain
\begin{align*}
\Tag{(1)}
\frac{\dd S}{\dd p}
  &= \frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
   + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p}
   + \frac{\dd R_{6}}{\dd p} + \text{etc.}, \\
\Tag{(2)}
\frac{\dd S}{\dd q}
  &= \frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
   + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q}
   + \frac{\dd R_{6}}{\dd q} + \text{etc.}, \\
\end{align*}
and therefore, by multiplying~(4) on \Pageref{58} by~(1), we obtain
\begin{multline*}
%[** TN: Re-broken]
\Tag{(3)}
\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p} \\
\begin{alignedat}{3}
  = p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
  &  + p\frac{\dd R_{4}}{\dd p}
  && + p\frac{\dd R_{5}}{\dd p}
  && + p\frac{\dd R_{6}}{\dd p} + \text{etc.} \\
%
  & - \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{4}}{\dd p} \\
%
 &&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p} \\
%
 &&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{3}}{\dd p} \\
%
&&&&&- (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\frac{\dd R_{2}}{\dd p} \\
&&&&&- \tfrac{7}{5}g'p^{2}q^{3}\frac{\dd R_{2}}{\dd p} \\
&&&&&- (\tfrac{3}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}\frac{\dd R_{2}}{\dd p},
\end{alignedat}
\end{multline*}
and multiplying~[3] on \Pageref{58} by~(2), we obtain
\begin{multline*}
%[** TN: Re-broken]
\Tag{(4)}
r\cos\psi·\frac{\dd S}{\dd q} \\
\begin{alignedat}{3}
  = q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
  &  + q\frac{\dd R_{4}}{\dd q}
  && + q\frac{\dd R_{5}}{\dd q}
  && + q\frac{\dd R_{6}}{\dd q} + \text{etc.} \\
%
  &+  \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{4}}{\dd q} \\
%
 &&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
  &&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{3}}{\dd q} \\
%
 &&&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q}
  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q} \\
%
&&&&&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\frac{\dd R_{2}}{\dd q} \\
&&&&&+ \tfrac{3}{5}g'p^{3}q^{2}\frac{\dd R_{2}}{\dd q} \\
&&&&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\frac{\dd R_{2}}{\dd q}.
\end{alignedat}
\end{multline*}
\PageSep{65}
Integrating~$n$ on \Pageref{36} with respect to~$q$, we find
\begin{alignat*}{2}
\Tag{(5)}
\int n\, dq = q + \tfrac{1}{2}f°q^{3}
  &+  \tfrac{1}{3}f'pq^{3}
  &&+ \tfrac{1}{3}f''p^{2}q^{3} + \text{etc.} \\
%
  &+  \tfrac{1}{4}g°q^{4}
  &&+ \tfrac{1}{4}g'pq^{4} + \text{etc.} \\
  &&&+\tfrac{1}{5}h°q^{5} + \text{etc.\ etc.}
\end{alignat*}
Multiplying~(4) on \Pageref{58} by~(5), we find
\begin{alignat*}{2}
\Tag{(6)}
\frac{r\sin\psi}{n}·\int n\, dq = pq - f°pq^{3}
  &- \tfrac{11}{12}f'p^{2}q^{3}
  &&- (\tfrac{13}{15}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{3} - \text{etc.} \\
%
  &-  \tfrac{5}{4}g°pq^{4}
  &&- \tfrac{23}{20}g'p^{2}q^{4} \\
  &&&- (\tfrac{7}{5}h° - \tfrac{16}{15}{f°}^{2})pq^{5}.
\end{alignat*}
Since
\[
\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p} + r\cos\psi·\frac{\dd S}{\dd q}
  = \frac{r\sin\psi}{n}·\int n\, dq,
\]
we obtain, by setting (6) equal to the sum of (3)~and~(4),
{\footnotesize
\begin{alignat*}{4}
&\Neg pq
&&- f°pq^{3}
&&- \tfrac{11}{12}f'p^{2}q^{3}
&&- (\tfrac{13}{15}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{3} - \text{etc.} \\
%
&&&&&- \tfrac{5}{4}g°pq^{4}
&&- \tfrac{23}{20}g'p^{2}q^{4} \\
%
&&&&&&&- (\tfrac{7}{5}h° - \tfrac{16}{15}{f°}^{2})pq^{5} \\
%%
&=  p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
&&+ p\frac{\dd R_{4}}{\dd p}
&&+ p\frac{\dd R_{5}}{\dd p} + q\frac{\dd R_{5}}{\dd q}
&&+ p\frac{\dd R_{6}}{\dd p}
  + \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q} + \text{etc.} \\
%
&+  q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
&&+ q\frac{\dd R_{4}}{\dd q}
&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
&&+ q\frac{\dd R_{6}}{\dd q}
  - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\frac{\dd R_{2}}{\dd p} \\
&&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{4}}{\dd p}
  + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\frac{\dd R_{2}}{\dd q} \\
%
&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{4}}{\dd q}
  - \tfrac{7}{5}g'p^{2}q^{3}\frac{\dd R_{2}}{\dd p} \\
%
&&&&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p}
  + \tfrac{3}{5}g'p^{3}q^{2}\frac{\dd R_{2}}{\dd q} \\
%
&&&&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{3}}{\dd q}
  - (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}\frac{\dd R_{2}}{\dd p} \\
%
&&&&&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q}
&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{3}}{\dd p}
  + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\frac{\dd R_{2}}{\dd q}.
\end{alignat*}}%
From this equation we find
\begin{align*}
R_{2} &= \tfrac{1}{2}pq,\quad
R_{3} = 0,\quad
R_{4} = -\tfrac{1}{12}f°pq^{3} - \tfrac{1}{12}f°p^{3}q, \\
%
R_{5} &= -\tfrac{1}{20}f'p^{4}q - \tfrac{3}{40}g°p^{3}q^{2}
  - \tfrac{7}{120}f'p^{2}q^{3} - \tfrac{1}{10}g°pq^{4}, \\
%
R_{6} &= -(\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}
  - (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
  &\quad- \tfrac{3}{40}g'p^{2}q^{4}
  - (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q
  - \tfrac{1}{20}g'p^{4}q^{2}.
\end{align*}
\PageSep{66}
Therefore we have
\begin{alignat*}{3}
\Tag{[7]}
S = \tfrac{1}{2}pq
  &-  \tfrac{1}{12}f°pq^{3}
  &&- \tfrac{1}{20}f'p^{4}q
  &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q - \text{etc.} \\
%
  &-  \tfrac{1}{12}f°p^{3}q
  &&- \tfrac{3}{40}g°p^{3}q^{2}
  &&- \tfrac{1}{20}g'p^{4}q^{2} \\
%
  &&&- \tfrac{7}{120}f'p^{2}q^{3}
  &&- (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
%
  &&&- \tfrac{1}{10}g°pq^{4}
  &&- \tfrac{3}{40}g'p^{2}q^{4} \\
%
  &&&&&- (\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}.
\end{alignat*}

\LineRef[25]{25}{Art.~25, p.~39, l.~17}. $3p^{2} + 4q^{2} + 4qq' + 4q'^{2}$ is replaced by $3p^{2} + 4q^{2} + 4q'^{2}$.
This error appears in all the reprints and translations (except Wangerin's).

\LineRef[26]{25}{Art.~25, p.~40, l.~8}. $3p^{2} - 2q^{2} + qq' + 4qq'$ is replaced by $3p^{2} - 2q^{2} + qq' + 4q'^{2}$.
This correction is noted in all the translations, and in Liouville's reprint.

\LineRef[27]{25}{Art.~25, p.~40}. Derivation of formulæ [8],~[9],~[10].

By priming the~$q$'s in~[7] we obtain at once a series for~$S'$. Then, since
$\sigma = S - S'$, we have
\begin{alignat*}{3}
\sigma = \tfrac{1}{2}p(q - q')
  &-  \tfrac{1}{12}f°p^{3}(q - q')
  &&- \tfrac{1}{20}f'p^{4}(q - q')
  &&- \tfrac{3}{40}g°p^{3}(q^{2} - q'^{2}) \\
%
  &-  \tfrac{1}{12}f°p(q^{3} - q'^{3})
  &&- \tfrac{7}{120}f'p^{2}(q^{3} - q'^{3})
  &&- \tfrac{1}{10}g°p(q^{4} - q'^{4}),
\end{alignat*}
correct to terms of the sixth degree.
%[** TN: Omitted line break in the original]
This expression may be written as follows:
\begin{align*}
\sigma = \tfrac{1}{2}p(q - q')
  \bigl(1 &- \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2}) \\
  &- \tfrac{1}{60}f'p(6p^{2} + 7q^{2} + 7qq' + 7q'^{2}) \\
  &- \tfrac{1}{20}g°(q + q')(3p^{2} + 4q^{2} + 4q'^{2})\bigr),
\end{align*}
or, after factoring,
{\small
\begin{multline*}
\Tag{(1)}
\sigma = \tfrac{1}{2}p(q - q')
  \bigl(1 - \tfrac{1}{3}f°q^{2} - \tfrac{1}{4}f'pq^{2} - \tfrac{1}{2}g°q^{3}\bigr)
  \bigl(1 - \tfrac{1}{6}f°(p^{2} - q^{2} + qq' + q'^{2}) \\
  - \tfrac{1}{60}f'p(6p^{2} - 8q^{2} + 7qq' + 7q'^{2})
  - \tfrac{1}{20}g°(3p^{2}q + 3p^{2}q' - 6q^{3} + 4q^{2}q' + 4qq'^{2} + 4q'^{3})\bigr).
\end{multline*}}%
The last factor on the right in~(1) can be written, thus:
\begin{alignat*}{5}
\bigl(1 &- \tfrac{2}{120}f°(4p^{2})
  &&- \tfrac{2}{120}f°(3p^{2})
  &&- \tfrac{2}{120}f'p(6qq')
  &&- \tfrac{2}{120}f°(3p^{2})
  &&- \tfrac{2}{120}f'p(qq') \\
%
  &+  \tfrac{2}{120}f°(2q^{2})
  &&+ \tfrac{2}{120}f°(6q^{2})
  &&- \tfrac{2}{120}f'p(3q'^{2})
  &&+ \tfrac{2}{120}f°(2q^{2})
  &&- \tfrac{2}{120}f'p(4q'^{2}) \\
%
  &-  \tfrac{2}{120}f°(3qq')
  &&- \tfrac{2}{120}f°(6qq')
  &&- \tfrac{6}{120}g°q(3p^{2})
  &&- \tfrac{2}{120}f°(qq')
  &&- \tfrac{6}{120}g°q'(3p^{2}) \\
%
  &-  \tfrac{2}{120}f°(3q'^{2})
  &&- \tfrac{2}{120}f°(3q'^{2})
  &&+ \tfrac{6}{120}g°q(6q^{2})
  &&- \tfrac{2}{120}f°(4q'^{2})
  &&+ \tfrac{6}{120}g°q'(2q^{2}) \\
%
 &&&- \tfrac{2}{120}f'p(3p^{2})
  &&- \tfrac{6}{120}g°q(6qq')
  &&- \tfrac{2}{120}f'p(3p^{2})
  &&- \tfrac{6}{120}g°q'(qq') \\
%
 &&&+ \tfrac{2}{120}f'p(6q^{2})
  &&- \tfrac{6}{120}g°q(3q'^{2})
  &&+ \tfrac{2}{120}f'p(2q^{2})
  &&- \tfrac{6}{120}g°q'(4q'^{2})\bigr).
\end{alignat*}
We know, further, that
\begin{align*}
&\,k = -\frac{1}{n}·\frac{\dd^{2} n}{\dd q^{2}}
   = -2f - 6gq - (12h - 2f^{2})q^{2} - \text{etc.}, \\
\PageSep{67}
&\begin{alignedat}{4}
f &= f° &&+ f'p &&+ f''p^{2} &&+ \text{etc.}, \\
g &= g° &&+ g'p &&+ g''p^{2} &&+ \text{etc.}, \\
h &= h° &&+ h'p &&+ h''p^{2} &&+ \text{etc.}
\end{alignedat}
\end{align*}
Hence, substituting these values for $f$,~$g$, and~$h$ in~$k$, we have at~$B$ where $k = \beta$,
correct to terms of the third degree,
\begin{align*}
\beta &= -2f° - 2f'p - 6g°q - 2f''p^{2} - 6g'pq - (12h° - 2{f°}^{2})q^{2}.
\intertext{Likewise, remembering that $q$~becomes~$q'$ at~$C$, and that both $p$~and~$q$ vanish at~$A$,
we have}
\gamma &= -2f° - 2f'p - 6g°q' - 2f''p^{2} - 6g'pq' - (12h° - 2{f°}^{2})q'^{2}, \\
\alpha &= -2f°.
\end{align*}
And since $c\sin B = r\sin\psi$,
\[
c\sin B = p(1 - \tfrac{1}{3}f°q^{2} - \tfrac{1}{4}f'pq^{2}
              - \tfrac{1}{2}g°q^{3} - \text{etc.}).
\]

Now, if we substitute in~(1) $c\sin B$, $\alpha$,~$\beta$,~$\gamma$ for the series which they represent,
and $a$~for~$q - q'$, we obtain (still correct to terms of the sixth degree)
\begin{align*}
\sigma = \tfrac{1}{2}ac\sin B\bigl(1
  &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} +  3qq' + 3q'^{2}) \\
  &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} +  6qq' + 3q'^{2}) \\
  &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} +\Z qq' + 4q'^{2})\bigr).
\end{align*}
And if in this equation we replace $p$,~$q$,~$q'$ by $c\sin B$, $c\cos B$, $c\cos B - a$, respectively,
we shall have
\begin{align*}
\Tag{[8]}
\sigma = \tfrac{1}{2}ac\sin B\bigl(1
  &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \Z9ac\cos B) \\
  &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\
  &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \Z9ac\cos B)\bigr).
\end{align*}

By writing for $B$,~$\alpha$,~$\beta$,~$a$ in~[8], $A$,~$\beta$,~$\alpha$,~$b$ respectively, we obtain at once
formula~[9]. Likewise by writing for $B$,~$\beta$,~$\gamma$,~$c$ in~[8], $C$,~$\gamma$,~$\beta$,~$b$ respectively, we
obtain formula~[10]. Formulæ [9]~and~[10] can, of course, also be derived by the
method used to derive~[8].

\LineRef[28]{26}{Art.~26, p.~41, l.~11}. The right hand side of this equation should have the positive
sign. All the editions prior to Wangerin's have the incorrect sign.

\LineRef[29]{26}{Art.~26, p.~41}. Derivation of formula~[11].

We have
\begin{align*}
\Tag{(1)}
r^{2}
  &+ r'^{2} - (q - q')^{2} - 2r\cos\phi·r'\cos\phi' - 2r\sin\phi·r'\sin\phi' \\
  &= b^{2} + c^{2} - a^{2} - 2bc\cos(\phi - \phi') \\
  &= 2bc(\cos A^{*} - \cos A),
\end{align*}
since $b^{2} + c^{2} - a^{2} = 2bc\cos A^{*}$ and $\cos(\phi - \phi') = \cos A$.
\PageSep{68}

By priming the $q$'s in formulæ [1],~[4],~[5] we obtain at once series for~$r'^{2}$,
$r'\cos\phi'$, $r'\sin\phi'$. Hence we have series for all the terms in the above expression,
and also for the terms in the expression:
\[
\Tag{(2)}
r\sin\phi·r'\cos\phi' - r\cos\phi·r'\sin\phi' = bc\sin A,
\]
namely,
\begin{alignat*}{3}
\Tag{(3)}
r^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q^{2}
  &&+ \tfrac{1}{2}f'p^{3}q^{2}
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2} + \text{etc.} \\
%
  &+ q^{2}
  &&+ \tfrac{1}{2}g°p^{2}q^{3}
  &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4},
\displaybreak[1] \\
%
\Tag{(4)}
r'^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q'^{2}
  &&+ \tfrac{1}{2}f'p^{3}q'^{2}
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q'^{2} + \text{etc.} \\
%
  &+ q'^{2}
  &&+ \tfrac{1}{2}g°p^{2}q'^{3}
  &&+ \tfrac{2}{5}g'p^{3}q'^{3} \\
  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q'^{4},
\end{alignat*}
\[
\Tag{(5)}
-(q - q')^{2} = -q^{2} + 2qq' - q'^{2},
\]
\begin{alignat*}{3}
\Tag{(6)}
2r\cos\phi
  &= 2p + \tfrac{4}{3}f°pq^{2}
  &&+ \tfrac{10}{12}f'p^{2}q^{2}
  &&+ (\tfrac{6}{10}f'' - \tfrac{16}{45}{f°}^{2})p^{3}q^{2} + \text{etc.} \\
%
  &&&+ g°pq^{3}
  &&+ \tfrac{14}{20}g'p^{2}q^{3} \\
%
  &&&&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})pq^{4},
\displaybreak[1] \\
%
\Tag{(7)}
r'\cos\phi'
  &= p + \tfrac{2}{3}f°pq'^{2}
  &&+ \tfrac{5}{12}f'p^{2}q'^{2}
  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q'^{2} + \text{etc.} \\
%
  &&&+ \tfrac{1}{2}g°pq'^{3}
  &&+ \tfrac{7}{20}g'p^{2}q'^{3} \\
%
  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq'^{4},
\displaybreak[1] \\
%
\Tag{(8)}
2r\sin\phi
  &= 2q - \tfrac{2}{3}f°p^{2}q
  &&- \tfrac{2}{6}f'p^{3}q
  &&- (\tfrac{2}{10}f'' - \tfrac{14}{90}{f°}^{2})p^{4}q - \text{etc.} \\
%
  &&&- \tfrac{2}{4}g°p^{2}q^{2}
  &&- \tfrac{6}{20}g'p^{3}q^{2} \\
%
  &&&&&- (\tfrac{2}{5}h° + \tfrac{26}{90}{f°}^{2})p^{2}q^{3},
\displaybreak[1] \\
%
\Tag{(9)}
r'\sin\phi'
  &= q' - \tfrac{1}{3}f°p^{2}q'
  &&- \tfrac{1}{6}f'p^{3}q'
  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q' + \text{etc.} \\
%
  &&&- \tfrac{1}{4}g°p^{2}q'^{2}
  &&- \tfrac{3}{20}g'p^{3}q'^{2} \\
%
  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q'^{3}.
\end{alignat*}
By adding (3),~(4), and~(5), we obtain
\begin{multline*}
\Tag{(10)}
r^{2} + r'^{2} - (q - q')^{2} \\
\begin{alignedat}{3}
  &= 2p^{2} + \tfrac{2}{3}f°p^{2}(q^{2} + q'^{2})
  &&+ \tfrac{1}{2}f'p^{3}(q^{2} + q'^{2})
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}(q^{2} + q'^{2}) + \text{etc.} \\
  &+ 2qq'
  &&+ \tfrac{1}{2}g°p^{2}(q^{3} + q'^{3})
  &&+ \tfrac{2}{5}g'p^{3}(q^{3} + q'^{3}) \\
%
&&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} + q'^{4}).
\end{alignedat}
\end{multline*}
On multiplying (6) by~(7), we obtain
\begin{multline*}
\Tag{(11)}
2r\cos\phi·r'\cos\phi' \\
\begin{alignedat}{2}
  = 2p^{2} + \tfrac{4}{3}f°p^{2}(q^{2} + q'^{2})
  &+ \tfrac{5}{6}f'p^{3}(q^{2} + q'^{2})
  &&+ (\tfrac{3}{5}f'' - \tfrac{16}{45}{f°}^{2})p^{4}(q^{2} + q'^{2}) + \text{etc.} \\
%
  &+ g°p^{2}(q^{3} + q'^{3})
  &&+ \tfrac{7}{10}g'p^{3}(q^{3} + q'^{3}) \\
%
  &&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}(q^{4} + q'^{4}) \\
  &&&+ \tfrac{8}{9}{f°}^{2}p^{2}q^{2}q'^{2},
\end{alignedat}
\end{multline*}
\PageSep{69}
and multiplying (8) by~(9), we obtain
\begin{alignat*}{3}
\Tag{(12)}
&\quad 2r\sin\phi·r'\sin\phi' \\
&= 2qq' - \tfrac{4}{3}f°p^{2}qq'
&&- \tfrac{2}{3}f'p^{3}qq'
&&- (\tfrac{2}{5}f'' - \tfrac{24}{45}{f°}^{2})p^{4}qq' - \text{etc.} \\
%
&&&- \tfrac{1}{2}g°p^{2}qq'(q + q')
&&- \tfrac{3}{10}g'p^{3}qq'(q + q') \\
&&&&&- (\tfrac{2}{5}h° + \tfrac{13}{45}{f°}^{2})p^{2}qq'(q^{2} + q'^{2}).
\end{alignat*}
Hence by adding (11)~and~(12), we have
{\small
\begin{multline*}
\Tag{(13)}
2bc\cos A \\
\begin{alignedat}{3}
&= 2p^{2} &&+ \tfrac{4}{3}f°p^{2}(q^{2} + q'^{2})
&&+ \tfrac{1}{6}f'p^{3}(5q^{2} - 4qq' + 5q'^{2})
  - \tfrac{8}{45}{f°}^{2}p^{4}(2q^{2} + 2q'^{2} - 3qq') + \text{etc.} \\
%
&+ 2qq' &&- \tfrac{4}{3}f°p^{2}qq'
&&+ \tfrac{1}{2}g°p^{2}(2q^{3} + 2q'^{3} - q^{2}q' - qq'^{2})
\end{alignedat} \\
\begin{aligned}
&- \tfrac{1}{45}{f°}^{2}p^{2}(14q^{4} + 14q'^{4} + 13q^{3}q' + 13 qq'^{3} - 40q^{2}q'^{2}) \\
&+ \tfrac{1}{10}g'p^{3}(7q^{3} + 7q'^{3} - 3q^{2}q' - 3qq'^{2}) \\
&+ \tfrac{1}{5}f''p^{4}(3q^{2} + 3q'^{2} - 2qq') \\
&+ \tfrac{2}{5}h°p^{2}(2q^{4} + 2q'^{4} - q^{3}q' - qq'^{3}).
\end{aligned}
\end{multline*}}%
Therefore we have, by subtracting (13) from~(10),
{\small
\begin{multline*}
2bc(\cos A^{*} - \cos A) \\
\begin{aligned}
= -\tfrac{2}{3}f°p^{2}(q^{2} + q'^{2} - 2qq')
&- \tfrac{1}{3}f'p^{3}(q^{2} + q'^{2} - 2qq')
 + \tfrac{4}{15}{f°}^{2}p^{4}(q^{2} + q'^{2} - 2qq') - \text{etc.} \\
&- \tfrac{1}{2}g°p^{2}(q^{3} + q'^{3} - q^{2}q' - qq'^{2})
 - \tfrac{1}{5}f''p^{4}(q^{2} + q'^{2} - 2qq')
\end{aligned} \\
\begin{aligned}
&+ \tfrac{1}{45}{f°}^{2}p^{2}(7q^{4} + 7q'^{4} + 13q^{3}q'+ 13qq'^{3} - 40q^{2}q'^{2}) \\
&- \tfrac{2}{5}h°p^{2}(q^{4} + q'^{4} - q^{3}q' - qq'^{3}) \\
&- \tfrac{3}{10}g'p^{3}(q^{3} + q'^{3} - q^{2}q' - qq'^{2}),
\end{aligned}
\end{multline*}}%
which we can write thus:
\begin{multline*}
\Tag{(14)}
%[** TN: Re-broken, explicit narrowing]
\qquad
2bc(\cos A^{*} - \cos A) \\
\begin{aligned}
  = -2p^{2}(q - q')^{2}&
  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
&+ \tfrac{1}{5}h°(q^{2} + qq' + q'^{2})
 + \tfrac{3}{20}g'p(q + q') \\
&- \tfrac{2}{15}{f°}^{2}p^{2}
 - \tfrac{1}{90}{f°}^{2}(7q^{2} + 7q'^{2} + 27qq')\bigr),\qquad
\end{aligned}
\end{multline*}
correct to terms of the seventh degree.

If we multiply (7) by~[5] on \Pageref{37}, we obtain
\begin{alignat*}{3}
\Tag{(15)}
%[** TN: Re-broken]
r\sin\phi·r'\cos\phi'
= pq &+ \tfrac{2}{3}f°pqq'^{2}
  &&+ \tfrac{5}{12}f'p^{2}qq'^{2}
  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}qq'^{2} - \text{etc.} \\
%
  &- \tfrac{1}{3}f°p^{3}q
  &&+ \tfrac{1}{2}g°pqq'^{3}
  &&+ \tfrac{7}{20}g'p^{2}qq'^{3} \\
%
  &&&- \tfrac{1}{6}f'p^{4}q
  &&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pqq'^{4} \\
%
  &&&- \tfrac{1}{4}g°p^{3}q^{2}
  &&- \tfrac{2}{9}{f°}^{2}p^{3}qq'^{2} \\
%
  &&&&&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{5}q \\
  &&&&&- \tfrac{3}{20}g'p^{4}q^{2} \\
  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{3}q^{3}.
\end{alignat*}
\PageSep{70}
And multiplying (9) by formula~[4] on \Pageref{37}, we obtain
\begin{alignat*}{3}
\Tag{(16)}
%[** TN: Re-broken]
r\cos\phi·r'\sin\phi'
= pq' &- \tfrac{1}{3}f°p^{3}q'
  &&- \tfrac{1}{6}f'p^{4}q'
  &&- (\tfrac{1}{16}f'' - \tfrac{7}{90}{f°}^{2})p^{5}q' + \text{etc.} \\
%
  &+  \tfrac{2}{3}f°pq^{2}q'
  &&- \tfrac{1}{4}g°p^{3}q'^{2}
  &&- \tfrac{3}{20}g'p^{4}q'^{2} \\
%
  &&&+ \tfrac{5}{12}f'p^{2}q^{2}q'
  &&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{3}q'^{3} \\
%
  &&&&&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}q' \\
  &&&&&+ \tfrac{7}{20}g'p^{2}q^{3}q' \\
  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}q'.
\end{alignat*}
Therefore we have, by subtracting (16) from~(15),
\begin{multline*}
\Tag{(17)}
bc\sin A \\
\begin{alignedat}{3}
=p(q - q')\bigl(1 &- \tfrac{1}{3}f°p^{2}
  &&- \tfrac{5}{12}f'pqq'
  &&- (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{2}qq' \\
%
  &-  \tfrac{2}{3}f°qq'
  &&- \tfrac{1}{6}f'p^{3}
  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4} \\
%
  &&&- \tfrac{1}{2}g°qq'(q + q')
  &&- \tfrac{7}{20}g'p qq'(q + q') \\
%
  &&&- \tfrac{1}{4}g°p^{2}(q + q')
  &&- \tfrac{3}{20}g'p^{3}(q + q') \\
%
  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}(q^{2} + qq' + q'^{2}) \\
%
  &&&&&- (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})qq'(q^{2} + qq' + q'^{2}) \\
  &&&&&+ \tfrac{2}{9}{f°}^{2}p^{2}qq'\bigr),
\end{alignedat}
\end{multline*}
correct to terms of the seventh degree.

Let $A^{*} - A = \zeta$, whence $A^{*} = A + \zeta$, $\zeta$~being a magnitude of the second order.
Hence we have, expanding $\sin\zeta$~and~$\cos\zeta$, and rejecting powers of~$\zeta$ above the second,
\[
\cos A^{*} = \cos A·\left(1 - \frac{\zeta^{2}}{2}\right) - \sin A·\zeta,
\]
or
\[
\cos A^{*} - \cos A = -\frac{\cos A}{2}·\zeta^{2} - \sin A·\zeta;
\]
or, multiplying both members of this equation by~$2bc$,
\[
\Tag{(18)}
2bc(\cos A^{*} - \cos A) = -bc\cos A·\zeta^{2} - 2bc\sin A·\zeta.
\]
Further, let $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, where the~$R$'s have the same meaning as before.
If now we substitute in~(18) for its various terms the series derived above, we shall
have, on rejecting terms above the sixth degree,
\begin{multline*}
(p^{2} + qq')R_{2}^{2} + 2p(q - q')
  \bigl(1 - \tfrac{1}{3}f°(p^{2} + 2qq')\bigr)\bigl(R_{2} + R_{3} + R_{4}\bigr) \\
  = 2p^{2}(q - q')^{2}
  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q') \\
  + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2})
  - \tfrac{1}{90}{f°}^{2}(12p^{2} + 7q^{2} + 7q'^{2} + 27qq')\bigr).)
\end{multline*}
\PageSep{71}
Equating terms of like powers, and solving for $R_{2}$,~$R_{3}$,~$R_{4}$, we find
\begin{align*}
R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
R_{3} = p(q - q')\bigl(\tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q')\bigr), \\
R_{4} &= p(q - q')\bigl(\tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q')
  + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
  &\qquad- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 7q'^{2} + 12qq')\bigr).
\end{align*}
Therefore we have
\begin{align*}
A^{*} - A = p(q - q')&\bigl(\tfrac{1}{3}{f°}^{2} + \tfrac{1}{6}f'p
  + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
  &+ \tfrac{3}{20}g'p(q + q') + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
  &- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 12qq' + 7q'^{2})\bigr),
\end{align*}
correct\Typo{,}{ to} terms of the fifth degree.

This equation may be written as follows:
\begin{align*}
A^{*} &= A + ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + q'^{2} + qq')\bigr)
  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') \\
  &+ \tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q')
   + \tfrac{1}{5}h°(q^{2} + qq' +q'^{2})
   - \tfrac{1}{90}{f°}^{2}(2p^{2} + 2q^{2} + 7qq' + 2q'^{2})\bigr).
\end{align*}
But, since
\[
2\sigma
  = ap\bigl(1 - \tfrac{1}{6}f°(p^{2}+ q^{2} + qq' + q'^{2}) + \text{etc.}\bigr),
\]
the above equation becomes
\begin{align*}
A^{*} = A - \sigma&\bigl(-\tfrac{2}{3}f° - \tfrac{1}{3}f'p
  - \tfrac{1}{2}g°(q + q') - \tfrac{1}{5}f''p^{2}
  - \tfrac{3}{10}g'p(q + q') \\
  &- \tfrac{2}{5}h°(q^{2} + qq' + q'^{2})
   + \tfrac{1}{90}{f°}^{2}(4p^{2} + 4q^{2} + 14qq' + 4q'^{2})\bigr),
\end{align*}
or
\begin{alignat*}{3}
A^{*} = A - \sigma\bigl(-\tfrac{2}{6}f°
  &- \tfrac{2}{12}f° &&- \tfrac{2}{12}f° \\
  &- \tfrac{2}{12}f'p   &&- \tfrac{2}{12}f'p \\
  &- \tfrac{6}{12}g°q &&- \tfrac{6}{12}g°q' \\
%
  &-  \tfrac{2}{12}f''p^{2}
  &&- \tfrac{2}{12}f''p^{2}
  &&+ \tfrac{2}{15}f''p^{2} \\
%
  &-  \tfrac{6}{12}g'pq
  &&- \tfrac{6}{12}g'pq'
  &&+ \tfrac{1}{5}g'p(q + q') \\
%
  &-  \tfrac{12}{12}h°q^{2}
  &&- \tfrac{12}{12}h°q'^{2}
  &&+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2}) \\
%
  &+  \tfrac{2}{12}{f°}^{2}q^{2}
  &&+ \tfrac{2}{12}{f°}^{2}q'^{2}
  &&+ \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr).
\end{alignat*}
Therefore, if we substitute in this equation $\alpha$,~$\beta$,~$\gamma$ for the series which they represent,
we shall have
\begin{align*}
\Tag{[11]}
A^{*} = A - \sigma
  &\bigl(\tfrac{1}{6}\alpha + \tfrac{1}{12}\beta + \tfrac{1}{12}\gamma
  + \tfrac{2}{15}f''p^{2} + \tfrac{1}{5}g'p(q + q') \\
  &+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2})
   + \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr).
\end{align*}

\LineRef[30]{26}{Art.~26, p.~41}. Derivation of formula~[12].

We form the expressions $(q - q')^{2} + r^{2} - r'^{2} - 2(q - q')r\cos\psi$ and $(q - q')r\sin\psi$.
Then, since
\begin{align*}
(q - q')^{2} + r^{2} - r'^{2} &= a^{2} + c^{2} - b^{2} = 2ac\cos B^{*}, \\
2(q - q')r\cos\psi &= 2ac\cos B,
\end{align*}
\PageSep{72}
we have
\[
(q - q')^{2} + r^{2} - r'^{2} - 2(q - q')r\cos\psi
  = 2ac(\cos B^{*} - \cos B).
\]
We have also
\[
(q - q')r\sin\psi = ac\sin B.
\]

Subtracting~(4) on \Pageref{68} from~[1] on \Pageref{36}, and adding this difference to
$(q - q')^{2}$, we obtain
\begin{multline*}
\Tag{(1)}
(q - q')^{2} + r^{2} - r'^{2},\quad\text{or}\quad 2ac\cos B^{*} \\
\begin{alignedat}{2}
= 2q(q - q') + \tfrac{2}{3}f°p^{2}(q^{2} - q'^{2})
  &+ \tfrac{1}{2}f'p^{3}(q^{2} - q'^{2})
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}(q^{2} - q'^{2})
    + \text{etc.} \\
%
  &+ \tfrac{1}{2}g°p^{2}(q^{3} - q'^{3})
  &&+ \tfrac{2}{5}g'p^{3}(q^{3} - q'^{3}) \\
%
  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} - q'^{4}).
\end{alignedat}
\end{multline*}
If we multiply~[3] on \Pageref{37} by~$2(q - q')$, we obtain
\begin{multline*}
\Tag{(2)}
2(q - q')r\cos\psi,\quad\text{or}\quad 2ac\cos B \\
\begin{alignedat}{2}
= 2q(q - q') + \tfrac{4}{3}f°p^{2}q(q - q')
  &+ f'p^{3}q(q - q')
  &&+ (\tfrac{4}{5}f'' - \tfrac{8}{45}{f°}^{2})p^{4}q(q - q') + \text{etc.} \\
%
  &+ \tfrac{3}{2}g°p^{2}q^{2}(q - q')
  &&+ \tfrac{6}{5}g'p^{3}q^{2}(q - q') \\
%
  &&&+ (\tfrac{8}{5}h° - \tfrac{28}{45}{f°}^{2})p^{2}q^{3}(q - q').
\end{alignedat}
\end{multline*}
Subtracting (2) from~(1), we have
\begin{multline*}
\Tag{(3)}
2ac(\cos B^{*} - \cos B) \\
\begin{alignedat}{2}
= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
  &+ \tfrac{1}{4}f'p
  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} + \text{etc.} \\
%
  &+ \tfrac{1}{4}g°(2q + q')
  &&+ \tfrac{1}{5}g'p(2q + q') \\
%
  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{20}{f°}^{2})
       (3q^{2} + 2qq' + q'^{2})\bigr).
\end{alignedat}
\end{multline*}

Multiplying [2] on \Pageref{36} by~$(q - q')$, we obtain at once
\begin{multline*}
\Tag{(4)}
(q - q')r\sin\psi,\quad\text{or}\quad ac\sin B \\
\begin{alignedat}{2}
= p(q - q')\bigl(1 - \tfrac{1}{3}f°q^{2}
  &- \tfrac{1}{4}f'pq^{2}
  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q^{2} + \text{etc.} \\
%
  &- \tfrac{1}{2}g°q^{3}
  &&- \tfrac{2}{5}g'pq^{3} \\
%
  &&&- (\tfrac{2}{5}h° - \tfrac{8}{45}{f°}^{2})q^{4}\bigr).
\end{alignedat}
\end{multline*}

We now set $B^{*} - B = \zeta$, whence $B^{*} = B + \zeta$, and therefore
\[
\cos B^{*} = \cos B \cos\zeta - \sin B \sin\zeta.
\]
This becomes, after expanding $\cos\zeta$ and $\sin\zeta$ and neglecting powers of~$\zeta$ above the
second,
\[
\cos B^{*} - \cos B = -\frac{\cos B}{2}·\zeta^{2} - \sin B·\zeta.
\]
Multiplying both members of this equation by~$2ac$, we obtain
\[
\Tag{(5)}
2ac(\cos B^{*} - \cos B) = -ac\cos B·\zeta^{2} - 2ac\sin B·\zeta.
\]
\PageSep{73}
Again, let $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, where the $R$'s have the same meaning as before.
Hence, replacing the terms in~(5) by the proper series and neglecting terms above the
sixth degree, we have
\begin{multline*}
\Tag{(6)}
q(q - q')R_{2}^{2} + 2p(q - q')(1 - \tfrac{1}{3}f°q^{2})
  (R_{2} + R_{3} - R_{4}) \\
\begin{alignedat}{2}
= 2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
  &+  \tfrac{1}{4}f'p
  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} \\
%
  &+ \tfrac{1}{4}g°(2q + q')
  &&+ \tfrac{1}{5}g'p(2q + q') \\
%
  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})(3q^{2} + 2qq' + q'^{2})\bigr).
\end{alignedat}
\end{multline*}
From this equation we find
\begin{align*}
R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
R_{3} = p(q - q')\bigl(\tfrac{1}{4}f'p + \tfrac{1}{4}g°(2q + q')\bigr), \\
R_{4} &= p(q - q')\bigl(\tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(2q + q')
  + \tfrac{1}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
&\qquad- \tfrac{1}{90}{f°}^{2}(4p^{2} + 16q^{2} + 9qq' + 7q'^{2})\bigr).
\end{align*}
Therefore we have, correct to terms of the fifth degree,
\begin{alignat*}{2}
B^{*} - B = p(q - q')\bigl(\tfrac{1}{3}f°
  &+  \tfrac{1}{4}f'p
  &&+ \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(2q + q') \\
%
  &+  \tfrac{1}{4}g°(2q + q')
  &&+ \tfrac{1}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
%
  &&&- \tfrac{1}{90}{f°}^{2}(4p^{2} + 16q^{2} + 9qq' + 7q'^{2})\bigr),
\end{alignat*}
or, after factoring the last factor on the right,
\begin{multline*}
\Tag{(7)}
%[** TN: Squeeze line so tag doesn't get pushed up]
\scalebox{0.975}[1]{$B^{*} - B - \tfrac{1}{2}p(q - q')
  \bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)
  \bigl(-\tfrac{2}{3}f° - \tfrac{1}{2}f'p - \tfrac{1}{2}g°(2q + q')$} \\
  -\tfrac{2}{5}f''p^{2} - \tfrac{2}{5}g'p(2q + q')
  - \tfrac{2}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
  - \tfrac{1}{90}{f°}^{2}(-2p^{2} + 22q^{2} + 8qq' + 4q'^{2})\bigr).
\end{multline*}
The last factor on the right in~(7) may be put in the form:
\begin{alignat*}{3}
\bigl(-\tfrac{2}{12}f°
  &-  \tfrac{2}{6}f°
  &&- \tfrac{2}{6}f° \\
%
  &-  \tfrac{2}{6}f'p
  &&- \tfrac{2}{12}f'p \\
%
  &-  \tfrac{6}{6}g°q
  &&- \tfrac{6}{12}g°q' \\
%
  &-  \tfrac{2}{6}f''p^{2}
  &&- \tfrac{2}{12}f''p^{2}
  &&+ \tfrac{1}{10}f''p^{2} \\
%
  &-  \tfrac{6}{6}g'pq
  &&- \tfrac{6}{12}g'pq'
  &&+ \tfrac{1}{10}g'p(2q + q') \\
%
  &-  \tfrac{12}{6}h°q^{2}
  &&- \tfrac{12}{12}h°q'^{2}
  &&+ \tfrac{1}{5}h°(4q^{2} + 3q'^{2} - 4qq') \\
%
  &+  \tfrac{2}{6}{f°}^{2}q^{2}
  &&+ \tfrac{2}{12}{f°}^{2}q'^{2}
  && - \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} + 11q'^{2} - 8qq')\bigr).
\end{alignat*}
Finally, substituting in~(7) $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions which they represent, we
obtain, still correct to terms of the fifth degree,
\begin{align*}
\Tag{[12]}
B^{*} = B - \sigma&\bigl(\tfrac{1}{12}\alpha + \tfrac{1}{6}\beta
                      + \tfrac{1}{12}\gamma + \tfrac{1}{10}f''p^{2} \\
  &+ \tfrac{1}{10}g'p(2q + q')
   + \tfrac{1}{5}h°(4q^{2} - 4qq' + 3q'^{2}) \\
  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} - 8qq' + 11q'^{2})\bigr).
\end{align*}
\PageSep{74}

\LineRef[30]{26}{Art.~26, p.~41}. Derivation of formula~[13].

Here we form the expressions $(q - q')^{2} + r'^{2} - r^{2} - 2(q - q')r'\cos(\pi - \psi')$ and
$(q - q')r'\sin(\pi - \psi')$ and expand them into series. Since
\begin{gather*}
(q - q')^{2} + r'^{2} - r^{2} = a^{2} + b^{2} - c^{2} = 2ab\cos C^{*}, \\
2(q - q')r'\cos(\pi - \psi') = 2ab\cos C,
\end{gather*}
we have
\[
(q - q')^{2} + r'^{2} - r^{2} - 2(q - q')r'\cos(\pi - \psi')
  = 2ab(\cos C^{*} - \cos C).
\]
We have also
\[
(q - q')r'\sin(\pi - \psi') = ab\sin C.
\]

Subtracting~(3) on \Pageref{68} from (4) on the same page, and adding the result to
$(q - q')^{2}$, we find
{\small
\begin{multline*}
\Tag{(1)}
(q - q')^{2} + r'^{2} - r^{2},\quad\text{or}\quad 2ab\cos C^{*} \\
\begin{alignedat}{2}
= -2q'(q - q') - \tfrac{2}{3}f°p^{2}(q^{2} - q'^{2})
  &- \tfrac{1}{2}f'p^{3}(q^{2} - q'^{2})
  &&- (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2}))p^{4}(q^{2} - q'^{2})
    - \text{etc.} \\
%
  &- \tfrac{1}{2}g°p^{2}(q^{3} - q'^{3})
  &&- \tfrac{2}{5}g'p^{3}(q^{3} - q'^{3}) \\
%
  &&&- (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} - q'^{4}).
\end{alignedat}
\end{multline*}}%
By priming the $q$'s in formula~[3] on \Pageref{37}, we get a series for $r\cos\psi'$, or for
$-r'\cos(\pi - \psi')$. If we multiply this series for $-r'\cos(\pi - \psi')$ by $2(q - q')$, we find
\begin{multline*}
\Tag{(2)}
-2(q - q')r'\cos(\pi - \psi'),\quad\text{or}\quad -2ab\cos C \\
\begin{alignedat}{2}
= 2(q - q')\bigl(q' + \tfrac{2}{3}f°p^{2}q'
  &+ \tfrac{1}{2}fp^{3}q'
  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q' + \text{etc.} \\
%
  &+  \tfrac{3}{4}g°p^{2}q'^{2}
  &&+ \tfrac{3}{5}g'p^{3}q'^{2} \\
%
  &&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q'^{3}\bigr).
\end{alignedat}
\end{multline*}
And therefore, by adding (1)~and~(2), we obtain
\begin{multline*}
\Tag{(3)}
2ab(\cos C^{*} - \cos C) \\
\begin{alignedat}{2}
= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°q'^{2}
  &+ \tfrac{1}{4}f'p
  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} + \text{etc.} \\
%
  &+ \tfrac{1}{4}g°(q + 2q')
  &&+ \tfrac{1}{5}g'p(q + 2q') \\
%
  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})
       (q^{2} + 2qq' + 3q'^{2})\bigr).
\end{alignedat}
\end{multline*}

By priming the $q$'s in~[2] on \Pageref{36}, we obtain a series for $r'\sin\psi'$, or for
$r'\sin(\pi - \psi')$. Then, multiplying this series for $r'\sin(\pi - \psi')$ by $(q - q')$, we find
\begin{multline*}
\Tag{(4)}
(q - q') r'\sin(\pi - \psi'),\quad\text{or}\quad ab\sin C \\
\begin{alignedat}{2}
= p(q - q')\bigl(1 - \tfrac{1}{3}f°q'^{2}
  &- \tfrac{1}{4}f'pq'^{2}
  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q'^{2} - \text{etc.} \\
%
  &- \tfrac{1}{2}g°q'^{3}
  &&- \tfrac{2}{5}g'pq'^{3} \\
%
  &&&- (\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})q'^{4}\bigr).
\end{alignedat}
\end{multline*}

As before, let $C^{*} - C = \zeta$, whence $C^{*} = C + \zeta$, and therefore
\[
\cos C^{*} = \cos C \cos\zeta - \sin C \sin \zeta.
\]
\PageSep{75}
Expanding $\cos\zeta$ and $\sin\zeta$ and neglecting powers of~$\zeta$ above the second, this equation
becomes
\[
\cos C^{*} - \cos C = -\frac{\cos C}{2}·\zeta^{2} - \sin C·\zeta,
\]
or, after multiplying both members by~$2ab$,
\[
\Tag{(5)}
2ab(\cos C^{*} - \cos C) = -ab\cos C·\zeta^{2} - 2ab\sin C·\zeta.
\]
Again we put $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, the $R$'s having the same meaning as before.
Now, by substituting (2),~(3),~(4) in~(5), and omitting terms above the sixth degree,
we obtain
\begin{multline*}
q'(q - q')R_{2}^{2}
  - 2p(q - q')\bigl(1 - \tfrac{1}{3}f°q'^{2})(R_{2} + R_{3} + R_{4}) \\
\begin{alignedat}{2}
= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
  &+ \tfrac{1}{4}f'p
  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} \\
%
  &+ \tfrac{1}{4}g°(q + 2q')
  &&+ \tfrac{1}{5}g'p(q + 2q') \\
%
  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})
       (q^{2} + 2qq' + 3q'^{2})\bigr),
\end{alignedat}
\end{multline*}
from which we find
\begin{align*}
R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
R_{3} = p(q - q')\bigl(\tfrac{1}{4}f'p + \tfrac{1}{4}g°(q + 2q')\bigr), \\
R_{4} &= p(q - q')\bigl(\tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q')
  + \tfrac{1}{5}h°(q^{2} + 2qq' + 3q'^{2}) \\
&\qquad- \tfrac{1}{90}{f°}^{2}(4p^{2} + 7q^{2} + 9qq' + 16q'^{2})\bigr).
\end{align*}
Therefore we have, correct to terms of the fifth degree,
\begin{alignat*}{2}
\Tag{(6)}
C^{*} - C = p(q - q')\bigl(\tfrac{1}{3}f°
  &+ \tfrac{1}{4}f'p
  &&+ \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q') \\
%
  &+ \tfrac{1}{4}g°(q + 2q')
  &&+ \tfrac{1}{5}h°(q^{2} + 2qq' + 3q'^{2}) \\
%
  &&&- \tfrac{1}{90}{f°}^{2}(4p^{2} + 7q^{2} + 9qq' + 16q'^{2})\bigr).
\end{alignat*}
The last factor on the right in~(6) may be written as the product of two factors, one
of which is $\frac{1}{2}\bigl(1 -\frac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)$, and the other,
\begin{align*}
2\bigl(\tfrac{1}{3}f° + \tfrac{1}{4}f'p
  &+ \tfrac{1}{4}g°(q + 2q') + \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q') \\
  &+ \tfrac{1}{5}h°(q^{2} + 3q'^{2} + 2qq')
   - \tfrac{1}{90}{f°}^{2}(-p^{2} + 2q^{2} + 4qq' + 11q'^{2})\bigr),
\end{align*}
or, in another form,
\begin{alignat*}{3}
-\bigl(-\tfrac{2}{12}f°
  &-  \tfrac{2}{12}f°
  &&- \tfrac{2}{6}f° \\
%
  &-  \tfrac{2}{12}f'p
  &&- \tfrac{2}{6}f'p \\
%
  &-  \tfrac{6}{12}g°q
  &&- \tfrac{6}{6}g°q' \\
%
  &-  \tfrac{2}{12}f''p^{2}
  &&- \tfrac{2}{6}f''p^{2}
  &&+ \tfrac{1}{10}f''p^{2} \\
%
  &-  \tfrac{6}{12}g'pq
  &&- \tfrac{6}{6}g'pq'
  &&+ \tfrac{1}{10}g'p(q + 2q') \\
%
  &-  \tfrac{12}{12}h°q^{2}
  &&- \tfrac{12}{6}h°q'^{2}
  &&+ \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
%
  &+ \tfrac{2}{12}{f°}^{2}q^{2}
  &&+ \tfrac{2}{6}{f°}^{2}q'^{2}
  &&- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr).
\end{alignat*}
\PageSep{76}
Hence (6) becomes, on substituting $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions which they represent,
\begin{align*}
\Tag{[13]}
C^{*} = C - \sigma
  &\bigl(\tfrac{1}{12}\alpha + \tfrac{1}{12}\beta + \tfrac{1}{6}\gamma
    + \tfrac{1}{10}f''p^{2} \\
  &+ \tfrac{1}{10}g'p(q + 2q') + \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr).
\end{align*}

\LineRef[31]{26}{Art.~26, p.~41}. Derivation of formula~[14].

This formula is derived at once by adding formulæ [11],~[12],~[13]. But, as
Gauss suggests, it may also be derived from~[6], \Pageref[p.]{38}. By priming the $q$'s in~[6]
we obtain a series for~$(\psi' + \phi')$. Subtracting this series from~[6], and noting that
$\phi - \phi' + \psi + \pi - \psi' = A + B + C$, we have, correct\Typo{}{ to} terms of the fifth degree,
\begin{alignat*}{2}
\Tag{(1)}
A + B + C = \pi - p(q - q')\bigl(f°
  &+ \tfrac{2}{3}f'p
  \rlap{${} + \tfrac{1}{2}f''p^{2} + \tfrac{3}{4}g'p(q + q')$} \\
%
  &+g°(q + q')
  &&+ h°(q^{2} + qq' + q'^{2}) \\
%
  &&&- \tfrac{1}{6}{f°}^{2}(p^{2} + 2q^{2} + 2qq' + 2q'^{2})\bigr).
\end{alignat*}
The second term on the right in~(1) may be written
\begin{align*}
+ \tfrac{1}{2}ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)
  ·2\bigl(-f°
  &- \tfrac{2}{3}f'p - \tfrac{1}{2}f''p^{2} - \tfrac{3}{4}g'p(q + q') \\
  &- g°(q + q') - h°(q^{2} + qq' + q'^{2}) \\
  &\qquad+ \tfrac{1}{6}{f°}^{2}(\Typo{+}{}q^{2} + qq' + q'^{2})\bigr),
\end{align*}
of which the last factor may be thrown into the form:
\begin{alignat*}{3}
\bigl(-\tfrac{2}{3}f°
  &-  \tfrac{2}{3}f°
  &&- \tfrac{2}{3}f° \\
%
  &-  \tfrac{2}{3}f'p
  &&- \tfrac{2}{3}f'p \\
%
  &-  \tfrac{6}{3}g°q
  &&- \tfrac{6}{3}g°q' \\
%
  &-  \tfrac{2}{3}f''p^{2}
  &&- \tfrac{2}{3}f''p^{2}
  &&+ \tfrac{1}{3}f''p^{2} \\
%
  &-  \tfrac{6}{3}g'pq
  &&- \tfrac{6}{3}g'pq'
  &&+ \tfrac{1}{2}g'p(q + q') \\
%
  &-  \tfrac{12}{3}h°q^{2}
  &&- \tfrac{12}{3}h°q'^{2}
  &&+ 2h°(q^{2} + q'^{2} - qq') \\
%
  &+  \tfrac{2}{3}{f°}^{2}q^{2}
  &&+ \tfrac{2}{3}{f°}^{2}q'^{2}
  &&- \tfrac{1}{3}{f°}^{2}(q^{2} + q'^{2} - qq')\bigr).
\end{alignat*}
Hence, by substituting $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions they represent, (1)~becomes
\begin{align*}
\Tag{[14]}
A + B + C = \pi + \sigma
  &\bigl(\tfrac{1}{3}\alpha + \tfrac{1}{3}\beta + \tfrac{1}{3}\gamma
    + \tfrac{1}{3}f''p^{2} \\
  &+ \tfrac{1}{2}g'p(q + q')
   + (2h° - \tfrac{1}{3}{f°}^{2})
     (q^{2} - qq' + q'^{2})\bigr).
\end{align*}

\LineRef{27}{Art.~27, p.~42}. Omitting terms above the second degree, we have
\[
a^{2} = q^{2} - 2qq' + q'^{2},\quad
b^{2} = p^{2} + q'^{2},\quad
c^{2} = p^{2} + q^{2}.
\]

The expressions in the parentheses of the first set of formulæ for $A^{*}$,~$B^{*}$,~$C^{*}$
in \Art{27} may be arranged in the following manner:
\[
\begin{array}{*{9}{r@{\,}}}
&(&2p^{2} - & q^{2} + &4qq' - & q'^{2} = \bigl(& (p^{2} + q'^{2}) + & (p^{2} + q^{2}) - & 2(q^{2} - 2qq' + q'^{2})\bigr), \\
&(& p^{2} - &2q^{2} + &2qq' + & q'^{2} = \bigl(&2(p^{2} + q'^{2}) - & (p^{2} + q^{2}) - &  (q^{2} - 2qq' + q'^{2})\bigr), \\
&(& p^{2} + & q^{2} + &2qq' - &2q'^{2} = \bigl(&-(p^{2} + q'^{2}) + &2(p^{2} + q^{2}) - &  (q^{2} - 2qq' + q'^{2})\bigr).
\end{array}
\]
\PageSep{77}
Now substituting $a^{2}$,~$b^{2}$,~$c^{2}$ for $q^{2} - 2qq' + q'^{2})$, $(p^{2} + q'^{2})$, $(p^{2} + q^{2})$ respectively, and
changing the signs of both members of the last two of these equations, we have
\[
\begin{array}{*{7}{r@{\,}}}
 (&2p^{2} - & q^{2} + &4qq' - & q'^{2} = (b^{2} + & c^{2} - & 2a^{2}), \\
-(& p^{2} - &2q^{2} + &2qq' + & q'^{2} = (a^{2} + & c^{2} - & 2b^{2}), \\
-(& p^{2} + & q^{2} + &2qq' - &2q'^{2} = (a^{2} + & b^{2} - & 2c^{2}).
\end{array}
\]
And replacing the expressions in the parentheses in the first set of formulæ for
$A^{*}$,~$B^{*}$,~$C^{*}$ by their equivalents, we get the second set.

\LineRef{27}{Art.~27, p.~42}. $f° = -\dfrac{1}{2R^{2}}$, $f'' = 0$, etc., may obtained directly, without the
%[** TN: Macro expands to "Arts. 25 and 26"]
use of the general considerations of \Arts[ and ]{25}{26}, in the following way. In the
case of the sphere
\[
ds^{2} = \cos^{2}\left(\frac{q}{R}\right)·dp^{2} + dq^{2},
\]
hence
\[
n = \cos\left(\frac{q}{R}\right)
  = 1 - \frac{q^{2}}{2R^{2}} + \frac{q^{4}}{24R^{4}} - \text{etc.},
\]
\ie,
\[
f° = -\frac{1}{2R^{2}},\quad
h° = \frac{1}{24R^{4}},\quad
f' = g° = f'' = g' = 0.\qquad\rlap{[Wangerin.]}
\]

\LineRef{27}{Art.~27, p.~42, l.~16}. This theorem of Legendre is found in the Mémoires (Histoire)
de l'\Typo{Academie}{Académie} Royale de Paris, 1787, p.~358, and also in his \Title{Trigonometry},
Appendix,~§\;V\@. He states it as follows in his \textit{Trigonometry}:

\begin{Theorem}[]
The very slightly curved spherical triangle, whose angles are $A$,~$B$,~$C$ and whose sides
are $a$,~$b$,~$c$, always corresponds to a rectilinear triangle, whose sides $a$,~$b$,~$c$ are of the same
lengths, and whose opposite angles are $A - \tfrac{1}{3}e$, $B - \tfrac{1}{3}e$, $C - \tfrac{1}{3}e$, $e$~being the excess of the
sum of the angles in the given spherical triangle over two right angles.
\end{Theorem}

\LineRef{28}{Art.~28, p.~43, l.~7}. The sides of this triangle are Hohehagen-Brocken, Inselberg-Hohehagen,
Brocken-Inselberg, and their lengths are about $107$, $85$, $69$~kilometers
respectively, according to Wangerin.

\LineRef{29}{Art.~29, p.~43}. Derivation of the relation between $\sigma$~and~$\sigma^{*}$.

In \Art{28} we found the relation
\[
A^{*} = A - \tfrac{1}{12}\sigma(2\alpha + \beta + \gamma).
\]
Therefore
\[
\sin A^{*}
  = \sin A\cos\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)
  - \cos A\sin\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr),
\]
which, after expanding $\cos\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ and $\sin\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ and rejecting
powers of $\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ above the first, becomes
\PageSep{78}
\[
\Tag{(1)}
\sin A^{*} = \sin A
  - \cos A·\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr),
\]
correct to terms of the fourth degree.

But, since $\sigma$~and~$\sigma^{*}$ differ only by terms above the second degree, we may replace
in~(1) $\sigma$~by the value of~$\sigma^{*}$, $\tfrac{1}{2}bc\sin A^{*}$. We thus obtain, with equal exactness,
\[
\Tag{(2)}
\sin A = \sin A^{*}
  \bigl(1 + \tfrac{1}{24}bc\cos A·(2\alpha + \beta + \gamma)\bigr).
\]
Substituting this value for~$\sin A$ in~[9], \Pageref[p.]{40}, we have, correct to terms of the sixth
degree, the first formula for~$\sigma$ given in \Art{29}. Since $2bc\cos A^{*}$, or $b^{2} + c^{2} - a^{2}$,
differs from~$2bc\cos A$ only by terms above the second degree, we may replace $2bc\cos A$
in this formula for~$\sigma$ by $b^{2} + c^{2} - a^{2}$. Also $\sigma^{*} = \tfrac{1}{2}bc \sin A^{*}$. Hence, if we make
these substitutions in the first formula for~$\sigma$, we obtain the second formula for~$\sigma$
with the same exactness. In the case of a sphere, where $\alpha = \beta = \gamma$, the second
formula for~$\sigma$ reduces to the third.

When the surface is spherical, (2)~becomes
\[
\sin A = \sin A^{*}(1 + \frac{\alpha}{6}bc \cos A).
\]
And replacing $2bc\cos A$ in this equation by $(b^{2} + c^{2} - a^{2})$, we have
\[
\sin A = \sin A^{*}\bigl(1 + \frac{\alpha}{12}(b^{2} + c^{2} - a^{2})\bigr),
\]
or
\[
\frac{\sin A}{\sin A^{*}}
  = \bigl(1 + \frac{\alpha}{12}(b^{2} + c^{2} - a^{2})\bigr).
\]
And likewise we can find
\[
\frac{\sin B}{\sin B^{*}}
  = \bigl(1 + \frac{\alpha}{12}(a^{2} + c^{2} - b^{2})\bigr),\qquad
\frac{\sin C}{\sin C^{*}}
  = \bigl(1 + \frac{\alpha}{12}(a^{2} + b^{2} - c^{2})\bigr).
\]
Multiplying together the last three equations and rejecting the terms containing $\alpha^{2}$~and~$\alpha^{3}$,
we have
\[
1 + \frac{\alpha}{12}(a^{2} + b^{2} + c^{2})
  = \frac{\sin A·\sin B·\sin C}{\sin A^{*}·\sin B^{*}·\sin C^{*}}.
\]
Finally, taking the square root of both members of this equation, we have, with the
same exactness,
\[
\sigma = 1 + \frac{\alpha}{24}(a^{2} + b^{2} + c^{2})
  = \Sqrt{\frac{\sin A·\sin B·\sin C}{\sin A^{*}·\sin B^{*}·\sin C^{*}}}.
\]

The method here used to derive the last formula from the next to the last
formula of \Art{29} is taken from Wangerin.
\PageSep{79}


\Paper{1825}
\null\vfill
\begin{center}
\LARGE
NEUE \\[12pt]
ALLGEMEINE UNTERSUCHUNGEN \\[12pt]
{\small ÜBER} \\[12pt]
DIE KRUMMEN FLÄCHEN \\[12pt]
{\normalsize [1825]} \\[12pt]
{\footnotesize
PUBLISHED POSTHUMOUSLY IN GAUSS'S WORKS, VOL.~VIII, 1901. PAGES 408--443}
\end{center}
\vfill
\cleardoublepage
\PageSep{80}
%[Blank page]
\PageSep{81}


\PaperTitle{\LARGE NEW GENERAL INVESTIGATIONS \\
{\small OF} \\
CURVED SURFACES \\
{\normalsize [1825]}}

Although the real purpose of this work is the deduction of new theorems concerning
its subject, nevertheless we shall first develop what is already known, partly
for the sake of consistency and completeness, and partly because our method of treatment
is different from that which has been used heretofore. We shall even begin by
advancing certain properties concerning plane curves from the same principles.


\Article{1.}

In order to compare in a convenient manner the different directions of straight
lines in a plane with each other, we imagine a circle with unit radius described
in the plane about an arbitrary centre. The position of the radius of this circle,
drawn parallel to a straight line given in advance, represents then the position of that
line. And the angle which two straight lines make with each other is measured by
the angle between the two radii representing them, or by the arc included between
their extremities. Of course, where precise definition is necessary, it is specified at
the outset, for every straight line, in what sense it is regarded as drawn. Without
such a distinction the direction of a straight line would always correspond to two
opposite radii.


\Article{2.}

In the auxiliary circle we take an arbitrary radius as the first, or its terminal
point in the circumference as the origin, and determine the positive sense of measuring
the arcs from this point (whether from left to right or the contrary); in the
opposite direction the arcs are regarded then as negative. Thus every direction of a
straight line is expressed in degrees,~etc., or also by a number which expresses them
in parts of the radius.
\PageSep{82}

Such lines as differ in direction by~$360°$, or by a multiple of~$360°$, have, therefore,
precisely the same direction, and may, generally speaking, be regarded as the
same. However, in such cases where the manner of describing a variable angle is
taken into consideration, it may be necessary to distinguish carefully angles differing
by~$360°$.

If, for example, we have decided to measure the arcs from left to right, and if
to two straight lines $l$,~$l'$ correspond the two directions $L$,~$L'$, then $L' - L$ is the angle
between those two straight lines. And it is easily seen that, since $L' - L$ falls
between $-180°$~and~$+180°$, the positive or negative value indicates at once that $l'$~lies
on the right or the left of~$l$, as seen from the point of intersection. This will
be determined generally by the sign of~$\sin(L' - L)$.

If $aa'$~is a part of a curved line, and if to the tangents at $a$,~$a'$ correspond
respectively the directions $\alpha$,~$\alpha'$, by which letters shall be denoted also the corresponding
points on the auxiliary circles, and if $A$,~$A'$ be their distances along the arc
from the origin, then the magnitude of the arc~$\alpha\alpha'$ or $A' - A$ is called the \emph{amplitude}
of~$aa'$.

The comparison of the amplitude of the arc~$aa'$ with its length gives us the
notion of curvature. Let $l$~be any point on the arc~$aa'$, and let $\lambda$,~$\Lambda$ be the same
with reference to it that $\alpha$,~$A$ and $\alpha'$,~$A'$ are with reference to $a$~and~$a'$. If now
$\alpha\lambda$~or~$\Lambda - A$ be proportional to the part~$al$ of the arc, then we shall say that $aa'$~is
uniformly curved throughout its whole length, and we shall call
\[
\frac{\Lambda - A}{al}
\]
the measure of curvature, or simply the curvature. We easily see that this happens
only when $aa'$~is actually the arc of a circle, and that then, according to our definition,
its curvature will be~$±\dfrac{1}{r}$ if $r$~denotes the radius. Since we always regard $r$
as positive, the upper or the lower sign will hold according as the centre lies to the
right or to the left of the arc~$aa'$ ($a$~being regarded as the initial point, $a'$~as the
end point, and the directions on the auxiliary circle being measured from left to
right). Changing one of these conditions changes the sign, changing two restores it
again.

On the contrary, if $\Lambda - A$ be not proportional to~$al$, then we call the arc non-uniformly
curved and the quotient
\[
\frac{\Lambda - A}{al}
\]
\PageSep{83}
may then be called its mean curvature. Curvature, on the contrary, always presupposes
that the point is determined, and is defined as the mean curvature of an element
at this point; it is therefore equal to
\[
\frac{d\Lambda}{d\,al}.
\]
We see, therefore, that arc, amplitude, and curvature sustain a similar relation to each
other as time, motion, and velocity, or as volume, mass, and density. The reciprocal
of the curvature, namely,
\[
\frac{d\,al}{d\Lambda},
\]
is called the radius of curvature at the point~$l$. And, in keeping with the above
conventions, the curve at this point is called concave toward the right and convex
toward the left, if the value of the curvature or of the radius of curvature happens
to be positive; but, if it happens to be negative, the contrary is true.


\Article{3.}

If we refer the position of a point in the plane to two perpendicular axes of
coordinates to which correspond the directions $0$~and~$90°$, in such a manner that the
first coordinate represents the distance of the point from the second axis, measured in
the direction of the first axis; whereas the second coordinate represents the distance
from the first axis, measured in the direction of the second axis; if, further, the indeterminates
$x$,~$y$ represent the coordinates of a point on the curved line, $s$~the length
of the line measured from an arbitrary origin to this point, $\phi$~the direction of the
tangent at this point, and $r$~the radius of curvature; then we shall have
\begin{align*}
dx &= \cos\phi·ds, \\
dy &= \sin\phi·ds, \\
r &= \frac{ds}{d\phi}.
\end{align*}

If the nature of the curved line is defined by the equation $V = 0$, where $V$~is a
function of $x$,~$y$, and if we set
\[
dV = p\, dx + q\, dy,
\]
then on the curved line
\[
p\, dx + q\, dy = 0.
\]
Hence
\[
p\cos\phi + q\sin\phi = 0,
\]
\PageSep{84}
and therefore
\[
\tan\phi = -\frac{p}{q}.
\]
We have also
\[
\cos\phi·dp + \sin\phi·dq - (p\sin\phi - q\cos\phi)\, d\phi = 0.
\]
If, therefore, we set, according to a well known theorem,
\begin{align*}
dp &= P\, dx + Q\, dy, \\
dq &= Q\, dx + R\, dy,
\end{align*}
then we have\Note{32}
\[
(P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi)\, ds
  = (p\sin\phi - q\cos\phi)\, d\phi,\NoteMark
\]
therefore
\[
\frac{1}{r}
  = \frac{P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi}
         {p\sin\phi - q\cos\phi},
\]
or, since\Note{33}
\begin{gather*}
\cos\phi = \frac{\mp q}{\Sqrt{p^{2} + q^{2}}},\qquad
\sin\phi = \frac{±p}{\Sqrt{p^{2} + q^{2}}};\NoteMark \\
±\frac{1}{r} = \frac{Pq^{2} - 2Qpq + Rp^{2}}{(p^{2} + q^{2})^{3/2}}.
\end{gather*}


\Article{4.}

The ambiguous sign in the last formula might at first seem out of place, but
upon closer consideration it is found to be quite in order. In fact, since this expression
depends simply upon the partial differentials of~$V$, and since the function $V$~itself
merely defines the nature of the curve without at the same time fixing the sense in
which it is supposed to be described, the question, whether the curve is convex
toward the right or left, must remain undetermined until the sense is determined by
some other means. The case is similar in the determination of~$\phi$ by means of the
tangent, to single values of which correspond two angles differing by~$180°$. The
sense in which the curve is described can be specified in the following different ways.

\Par{I.} By means of the sign of the change in~$x$. If $x$~increases, then $\cos\phi$ must be
positive. Hence the upper signs will hold if $q$~has a negative value, and the lower
signs if $q$~has a positive value. When $x$~decreases, the contrary is true.

\Par{II.} By means of the sign of the change in~$y$. If $y$~increases, the upper signs
must be taken when $p$~is positive, the lower when $p$~is negative. The contrary is
true when $y$~decreases.

\Par{III.} By means of the sign of the value which the function~$V$ takes for points
not on the curve. Let $\delta x$,~$\delta y$ be the variations of $x$,~$y$ when we go out from the
\PageSep{85}
curve toward the right, at right angles to the tangent, that is, in the direction~$\phi + 90°$;
and let the length of this normal be~$\delta\rho$. Then, evidently, we have
\begin{align*}
\delta x &= \delta\rho·\cos(\phi + 90°), \\
\delta y &= \delta\rho·\sin(\phi + 90°),
\end{align*}
or
\begin{align*}
\delta x &= -\delta\rho·\sin\phi, \\
\delta y &= +\delta\rho·\cos\phi.
\end{align*}
Since now, when $\delta\rho$~is infinitely small,
\begin{align*}
\delta V &= p\, \delta x + q\, \delta y \\
  &= (-p\sin\phi + q\cos\phi)\, \delta\rho \\
  &= \mp\delta\rho\Sqrt{p^{2} + q^{2}}\Add{,}
\end{align*}
and since on the curve itself $V$~vanishes, the upper signs will hold if~$V$, on passing
through the curve from left to right, changes from positive to negative, and the contrary.
If we combine this with what is said at the end of \Art{2}, it follows that the
curve is always convex toward that side on which $V$~receives the same sign as
\[
Pq^{2} - 2Qpq + Rp^{2}.
\]

For example, if the curve is a circle, and if we set
\[
V = x^{2} + y^{2} - a^{2}\Add{,}
\]
then we have
\begin{gather*}
p = 2x,\qquad q = 2y, \\
P = 2,\qquad Q = 0,\qquad R = 2, \\
Pq^{2} - 2Qpq + Rp^{2} = 8y^{2} + 8x^{2} = 8a^{2}, \\
(p^{2} + q^{2})^{3/2} = 8a^{3}, \\
r = ± a\Add{;}
\end{gather*}
and the curve will be convex toward that side for which
\[
x^{2} + y^{2} > a^{2},
\]
as it should be.

The side toward which the curve is convex, or, what is the same thing, the signs
in the above formulæ, will remain unchanged by moving along the curve, so long as
\[
\frac{\delta V}{\delta\rho}
\]
does not change its sign. Since $V$~is a continuous function, such a change can take
place only when this ratio passes through the value zero. But this necessarily presupposes
that $p$~and~$q$ become zero at the same time. At such a point the radius
\PageSep{86}
of curvature becomes infinite or the curvature vanishes. Then, generally speaking,
since here
\[
-p\sin\phi + q\cos\phi
\]
will change its sign, we have here a point of inflexion.


\Article{5.}

The case where the nature of the curve is expressed by setting $y$~equal to a
given function of~$x$, namely, $y = X$, is included in the foregoing, if we set
\[
V = X - y.
\]
If we put
\[
dX = X'\, dx,\qquad
dX' = X''\, dx,
\]
then we have
\begin{gather*}
p = X',\qquad q = -1, \\
P = X'', \qquad Q = 0,\qquad R = 0,
\end{gather*}
therefore
\[
±\frac{1}{r} = \frac{X''}{(1 + X'^{2})^{3/2}}.
\]
Since $q$~is negative here, the upper sign holds for increasing values of~$x$. We can
therefore say, briefly, that for a positive~$X''$ the curve is concave toward the same
side toward which the $y$-axis lies with reference to the $x$-axis; while for a negative~$X''$
the curve is convex toward this side.


\Article{6.}

If we regard $x$,~$y$ as functions of~$s$, these formulæ become still more elegant.
Let us set
\begin{alignat*}{2}
\frac{dx}{ds} &= x',\qquad& \frac{dx'}{ds} &= x'', \\
\frac{dy}{ds} &= y',\qquad& \frac{dy'}{ds} &= y''.
\end{alignat*}
Then we shall have
\begin{alignat*}{2}
x'  &= \cos\phi,\qquad & y' &= \sin\phi, \\
x'' &= -\frac{\sin\phi}{r},\qquad & y'' &= \frac{\cos\phi}{r};
\intertext{or}
y' &= -rx'',\qquad& x' &= ry'',
\end{alignat*}
\PageSep{87}
or also
\[
1 = r(x'y'' - y'x''),
\]
so that
\[
x'y'' - y'x''
\]
represents the curvature, and
\[
\frac{1}{x'y'' - y'x''}
\]
the radius of curvature.


\Article{7.}

We shall now proceed to the consideration of curved surfaces. In order to represent
the directions of straight lines in space considered in its three dimensions, we
imagine a sphere of unit radius described about an arbitrary centre. Accordingly, a
point on this sphere will represent the direction of all straight lines parallel to the
radius whose extremity is at this point. As the positions of all points in space
are determined by the perpendicular distances $x$,~$y$,~$z$ from three mutually perpendicular
planes, the directions of the three principal axes, which are normal to these
principal planes, shall be represented on the auxiliary sphere by the three points
$(1)$,~$(2)$,~$(3)$. These points are, therefore, always $90°$~apart, and at once indicate the
sense in which the coordinates are supposed to increase. We shall here state several
well known theorems, of which constant use will be made.

\Par{1)} The angle between two intersecting straight lines is measured by the arc [of
the great circle] between the points on the sphere which represent their directions.

\Par{2)} The orientation of every plane can be represented on the sphere by means
of the great circle in which the sphere is cut by the plane through the centre parallel
to the first plane.

\Par{3)} The angle between two planes is equal to the angle between the great circles
which represent their orientations, and is therefore also measured by the angle
between the poles of the great circles.

\Par{4)} If $x$,~$y$,~$z$; $x'$,~$y'$,~$z'$ are the coordinates of two points, $r$~the distance between
them, and $L$~the point on the sphere which represents the direction of the straight
line drawn from the first point to the second, then
\begin{alignat*}{2}
x' &= x &&+ r\cos(1)L, \\
y' &= y &&+ r\cos(2)L, \\
2' &= z &&+ r\cos(3)L.
\end{alignat*}

\Par{5)} It follows immediately from this that we always have
\[
\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1
\]
\PageSep{88}
[and] also, if $L'$~is any other point on the sphere,
\[
\cos(1)L·\cos(1)L' + \cos(2)L·\cos(2)L' + \cos(3)L·\cos(3)L' = \cos LL'.
\]

We shall add here another theorem, which has appeared nowhere else, as far as
we know, and which can often be used with advantage.

Let $L$, $L'$, $L''$, $L'''$ be four points on the sphere, and $A$~the angle which $LL'''$
and $L'L''$ make at their point of intersection. [Then we have]
\[
\cos LL'·\cos L''L''' - \cos LL''·\cos L'L''' = \sin LL'''·\sin L'L''·\cos A.
\]

The proof is easily obtained in the following way. Let
\[
AL = t,\qquad
AL' = t',\qquad
AL'' = t'',\qquad
AL''' = t''';
\]
we have then
\begin{alignat*}{6}
&\cos L  L'   &&= \cos t  &&\cos t'   &&+ \sin t  &&\sin t'  &&\cos A, \\
&\cos L''L''' &&= \cos t''&&\cos t''' &&+ \sin t''&&\sin t'''&&\cos A, \\
&\cos L  L''  &&= \cos t  &&\cos t''  &&+ \sin t  &&\sin t'' &&\cos A, \\
&\cos L' L''' &&= \cos t' &&\cos t''' &&+ \sin t' &&\sin t'''&&\cos A.
\end{alignat*}
Therefore
\begin{multline*}
\cos LL' \cos L''L''' - \cos LL'' \cos L'L'' \\
\begin{aligned}
&= \cos A \{\cos t  \cos t'  \sin t''\sin t'''
          + \cos t''\cos t'''\sin t  \sin t' \\
&\qquad\qquad
  - \cos t\cos t''\sin t'\sin t''' - \cos t'\cos t'''\sin t\sin t''\} \\
&= \cos A (\cos t \sin t''' - \cos t'''\sin t)
          (\cos t'\sin t''  - \cos t'' \sin t') \\
&= \cos A \sin (t''' - t) \sin(t'' - t') \\
&= \cos A \sin LL''' \sin L'L''.
\end{aligned}
\end{multline*}

Since each of the two great circles goes out from~$A$ in two opposite directions,
two supplementary angles are formed at this point. But it is seen from our analysis
that those branches must be chosen, which go in the same sense from~$L$ toward~$L'''$
and from $L'$~toward~$L''$.

Instead of the angle~$A$, we can take also the distance of the pole of the great
circle~$LL'''$ from the pole of the great circle~$L'L''$. However, since every great circle
has two poles, we see that we must join those about which the great circles run in
the same sense from~$L$ toward~$L'''$ and from~$L'$ toward~$L''$, respectively.

The development of the special case, where one or both of the arcs $LL'''$~and~$L'L''$ are~$90°$, we leave to the reader.

\Par{6)} Another useful theorem is obtained from the following analysis. Let $L$,~$L'$,~$L''$
be three points upon the sphere and put
\PageSep{89}
\begin{alignat*}{6}
&\cos L  &&(1) = x,       &&\cos L   &&(2) = y,       &&\cos L   &&(3) = z, \\
&\cos L' &&(1) = x',      &&\cos L'  &&(2) = y',      &&\cos L'  &&(3) = z', \\
&\cos L''&&(1) = x'',\quad&&\cos L'' &&(2) = y'',\quad&&\cos L'' &&(3) = z''.
\end{alignat*}

We assume that the points are so arranged that they run around the triangle
included by them in the same sense as the points $(1)$,~$(2)$,~$(3)$. Further, let $\lambda$~be
that pole of the great circle~$L'L''$ which lies on the same side as~$L$. We then have,
from the above lemma,
\begin{alignat*}{3}
&y'z'' &&- z'y'' &&= \sin L'L''·\cos\lambda(1), \\
&z'x'' &&- x'z'' &&= \sin L'L''·\cos\lambda(2), \\
&x'y'' &&- y'x'' &&= \sin L'L''·\cos\lambda(3).
\end{alignat*}
Therefore, if we multiply these equations by $x$,~$y$,~$z$ respectively, and add the products,
we obtain\Note{34}
\[
xy'z'' + x'y''z + x''yz' - xy''z' - x'yz'' - x''y'z
  = \sin L'L''·\cos\lambda L,\NoteMark
\]
wherefore, we can write also, according to well known principles of spherical trigonometry,
\begin{alignat*}{2}
  \sin L'L''·&\sin L L''&&·\sin L' \\
= \sin L'L''·&\sin L L' &&·\sin L'' \\
= \sin L'L''·&\sin L'L''&&·\sin L,
\end{alignat*}
if $L$,~$L'$,~$L''$ denote the three angles of the spherical triangle. At the same time we
easily see that this value is one-sixth of the pyramid whose angular points are the
centre of the sphere and the three points $L$,~$L'$,~$L''$ (and indeed \emph{positive}, if~etc.).


\Article{8.}

The nature of a curved surface is defined by an equation between the coordinates
of its points, which we represent by
\[
f(x, y, z) = 0.\NoteMark
\]
Let the total differential of $f(x, y, z)$ be
\[
P\, dx + Q\, dy + R\, dz,
\]
where $P$,~$Q$,~$R$ are functions of $x$,~$y$,~$z$. We shall always distinguish two sides of the
surface, one of which we shall call the upper, and the other the lower. Generally
speaking, on passing through the surface the value of~$f$ changes its sign, so that, as
long as the continuity is not interrupted, the values are positive on one side and negative
on the other.
\PageSep{90}

The direction of the normal to the surface toward that side which we regard as
the upper side is represented upon the auxiliary sphere by the point~$L$. Let
\[
\cos L(1) = X,\qquad
\cos L(2) = Y,\qquad
\cos L(3) = Z.
\]
Also let $ds$~denote an infinitely small line upon the surface; and, as its direction is
denoted by the point~$\lambda$ on the sphere, let
\[
\cos \lambda(1) = \xi,\qquad
\cos \lambda(2) = \eta,\qquad
\cos \lambda(3) = \zeta.
\]
We then have
\[
dx = \xi\, ds,\qquad
dy = \eta\, ds,\qquad
dz = \zeta\, ds,
\]
therefore
\[
P\xi + Q\eta + R\zeta = 0,
\]
and, since $\lambda L$ must be equal to~$90°$, we have also
\[
X\xi + Y\eta + Z\zeta = 0.
\]
Since $P$,~$Q$,~$R$, $X$,~$Y$,~$Z$ depend only on the position of the surface on which we take
the element, and since these equations hold for every direction of the element on the
surface, it is easily seen that $P$,~$Q$,~$R$ must be proportional to $X$,~$Y$,~$Z$. Therefore
\[
P = X\mu,\qquad
Q = Y\mu,\qquad
R = Z\mu\Typo{,}{.}
\]
Therefore, since
\begin{gather*}
X^{2} + Y^{2} + Z^{2} = 1; \\
\mu = PX + QY + RZ
\intertext{and}
\mu^{2} = P^{2} + Q^{2} + R^{2}, \\
\intertext{or}
\mu = ±\Sqrt{P^{2} + Q^{2} + R^{2}}.
\end{gather*}

If we go out from the surface, in the direction of the normal, a distance equal to
the element~$\delta\rho$, then we shall have
\[
\delta x = X\, \delta\rho,\qquad
\delta y = Y\, \delta\rho,\qquad
\delta z = Z\, \delta\rho
\]
and
\[
\delta f = P\, \delta x + Q\, \delta y + R\, \delta z = \mu\, \delta\rho.
\]
We see, therefore, how the sign of~$\mu$ depends on the change of sign of the value of~$f$
in passing from the lower to the upper side.


\Article{9.}

Let us cut the curved surface by a plane through the point to which our notation
refers; then we obtain a plane curve of which $ds$~is an element, in connection
with which we shall retain the above notation. We shall regard as the upper side of
the plane that one on which the normal to the curved surface lies. Upon this plane
\PageSep{91}
we erect a normal whose direction is expressed by the point~$\L$ of the auxiliary
sphere. By moving along the curved line, $\lambda$~and~$L$ will therefore change their positions,
while $\L$~remains constant, and $\lambda L$~and~$\lambda\L$ are always equal to~$90°$. Therefore
$\lambda$~describes the great circle one of whose poles is~$\L$. The element of this great circle
will be equal to~$\dfrac{ds}{r}$, if $r$~denotes the radius of curvature of the curve. And again,
if we denote the direction of this element upon the sphere by~$\lambda'$, then $\lambda'$~will evidently
lie in the same great circle and be $90°$~from~$\lambda$ as well as from~$\L$. If we
now set
\[
\cos \lambda'(1) = \xi',\qquad
\cos \lambda'(2) = \eta',\qquad
\cos \lambda'(3) = \zeta',
\]
then we shall have
\[
d\xi = \xi'\, \frac{ds}{r},\qquad
d\eta = \eta'\, \frac{ds}{r},\qquad
d\zeta = \zeta'\, \frac{ds}{r},
\]
since, in fact, $\xi$,~$\eta$,~$\zeta$ are merely the coordinates of the point~$\lambda$ referred to the centre
of the sphere.

Since by the solution of the equation $f(x, y, z) = 0$ the coordinate~$z$ may be
expressed in the form of a function of $x$,~$y$, we shall, for greater simplicity, assume
that this has been done and that we have found
\[
z = F(x, y).
\]
We can then write as the equation of the surface
\[
z - F(x, y) = 0,
\]
or
\[
f(x, y, z) = z - F(x, y).
\]

From this follows, if we set
\begin{gather*}
dF(x, y) = t\, dx + u\, dy, \\
P = -t,\qquad
Q = -u,\qquad
R = 1,
\end{gather*}
where $t$,~$u$ are merely functions of $x$~and~$y$. We set also
\[
dt = T\, dx + U\, dy,\qquad
du = U\, dx + V\, dy.
\]

Therefore upon the whole surface we have
\[
dz = t\, dx + u\, dy
\]
and therefore, on the curve,
\[
\zeta = t\xi + u\eta.
\]
Hence differentiation gives, on substituting the above values for $d\xi$,~$d\eta$,~$d\zeta$,
\begin{align*}
(\zeta' - t\xi' - u\eta') \frac{ds}{r}
  &= \xi\, dt + \eta\, du \\
  &= (\xi^{2}T + 2\xi\eta U + \eta^{2}V)\, ds,
\end{align*}
\PageSep{92}
or
\begin{align*}
\frac{1}{r}
  &= \frac{\xi^{2}T + 2\xi\eta U + \eta^{2}V}{-\xi' t - \eta'\Typo{\mu}{u} + \zeta'} \\
  &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{X\xi' - Y\eta' + Z\zeta'} \\
  &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\cos L\lambda'}.
\end{align*}


\Article{10.}

Before we further transform the expression just found, we will make a few
remarks about it.

A normal to a curve in its plane corresponds to two directions upon the sphere,
according as we draw it on the one or the other side of the curve. The one direction,
toward which the curve is \emph{concave}, is denoted by~$\lambda'$, the other by the opposite
point on the sphere. Both these points, like $L$~and~$\L$, are $90°$~from~$\lambda$, and therefore
lie in a great circle. And since $\L$~is also $90°$~from~$\lambda$, $\L L = 90° - L\lambda'$, or
$= L\lambda' - 90°$. Therefore
\[
\cos L\lambda' = ±\sin \L L,
\]
where $\sin \L L$ is necessarily positive. Since $r$~is regarded as positive in our analysis,
the sign of~$\cos L\lambda'$ will be the same as that of
\[
Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).
\]
And therefore a positive value of this last expression means that $L\lambda'$~is less than~$90°$,
or that the curve is concave toward the side on which lies the projection of the
normal to the surface upon the plane. A negative value, on the contrary, shows that
the curve is convex toward this side. Therefore, in general, we may set also
\[
\frac{1}{r} = \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\sin \L L},
\]
if we regard the radius of curvature as positive in the first case, and negative in
the second. $\L L$~is here the angle which our cutting plane makes with the plane
tangent to the curved surface, and we see that in the different cutting planes passed
through the same point and the same tangent the radii of curvature are proportional
to the sine of the inclination. Because of this simple relation, we shall limit ourselves
hereafter to the case where this angle is a right angle, and where the cutting
\PageSep{93}
plane, therefore, is passed through the normal of the curved surface. Hence we have
for the radius of curvature the simple formula
\[
\frac{1}{r} = Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).
\]


\Article{11.}

Since an infinite number of planes may be passed through this normal, it follows
that there may be infinitely many different values of the radius of curvature. In this
case $T$,~$U$,~$V$,~$Z$ are regarded as constant, $\xi$,~$\eta$,~$\zeta$ as variable. In order to make the
latter depend upon a single variable, we take two fixed points $M$,~$M'$ $90°$~apart on the
great circle whose pole is~$L$. Let their coordinates referred to the centre of the sphere
be $\alpha$,~$\beta$,~$\gamma$;  $\alpha'$,~$\beta'$,~$\gamma'$. We have then
\[
\cos\lambda(1)
  = \cos\lambda M ·\cos M(1)
  + \cos\lambda M'·\cos M'(1)
  + \cos\lambda L ·\cos L(1).
\]
If we set
\[
\lambda M = \phi,
\]
then we have
\[
\cos\lambda M' = \sin\phi,
\]
and the formula becomes
\begin{align*}
\xi   &= \alpha\cos\phi + \alpha'\sin\phi,
\intertext{and likewise}
\eta  &= \beta \cos\phi + \beta' \sin\phi, \\
\zeta &= \gamma\cos\phi + \gamma'\sin\phi.
\end{align*}

Therefore, if we set\Note{35}
\begin{align*}
A &= (\alpha^{2}T + 2\alpha\beta U + \beta^{2}V)Z, \\
B &= (\alpha\alpha'T + (\alpha'\beta + \alpha\beta')U + \beta\beta'V)Z,\NoteMark \\
C &= (\alpha'^{2}T + 2\alpha'\beta' U + \beta'^{2}V)Z,
\end{align*}
we shall have
\begin{align*}
\frac{1}{r}
  &= A\cos^{2}\phi + 2B\cos\phi \sin\phi + C\sin^{2}\phi \\
  &= \frac{A + C}{2} + \frac{A - C}{2}\cos 2\phi + B\sin 2\phi.
\end{align*}
If we put
\begin{align*}
\frac{A - C}{2} &= E\cos 2\theta, \\
B &= E\sin 2\theta,
\end{align*}
\PageSep{94}
where we may assume that $E$~has the same sign as~$\dfrac{A - C}{2}$, then we have
\[
\frac{1}{r} = \tfrac{1}{2}(A + C) + E\cos 2(\phi - \theta).
\]
It is evident that $\phi$~denotes the angle between the cutting plane and another plane
through this normal and that tangent which corresponds to the direction~$M$. Evidently,
therefore, $\dfrac{1}{r}$~takes its greatest (absolute) value, or $r$~its smallest, when $\phi = \theta$; and $\dfrac{1}{r}$~its
smallest absolute value, when $\phi = \theta + 90°$. Therefore the greatest and the least
curvatures occur in two planes perpendicular to each other. Hence these extreme
values for~$\dfrac{1}{r}$ are
\[
\tfrac{1}{2}(A + C) ± \SQRT{\left(\frac{A - C}{2}\right)^{2} + B^{2}}.
\]
Their sum is $A + C$ and their product $AC - B^{2}$, or the product of the two extreme
radii of curvature is
\[
= \frac{1}{AC - B^{2}}.
\]
This product, which is of great importance, merits a more rigorous development.
In fact, from formulæ above we find
\[
AC - B^{2} = (\alpha\beta' -\beta\alpha')^{2}(TV - U^{2})Z^{2}.
\]
But from the third formula in [Theorem]~6, \Art{7}, we easily infer that\Note{36}
\[
\alpha\beta' - \beta\alpha' = ±Z,\NoteMark
\]
therefore
\[
AC - B^{2} = Z^{4}(TV - U^{2}).
\]
Besides, from \Art{8},
\begin{align*}
Z &= ±\frac{R}{\Sqrt{P^{2} + Q^{2} + R^{2}}} \\
  &= ±\frac{1}{\Sqrt{1 + t^{2} + u^{2}}},
\end{align*}
therefore
\[
AC - B^{2} = \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}.
\]

Just as to \emph{each} point on the curved surface corresponds a particular point~$L$ on
the auxiliary sphere, by means of the normal erected at this point and the radius of
\PageSep{95}
the auxiliary sphere parallel to the normal, so the aggregate of the points on the
auxiliary sphere, which correspond to all the points of a \emph{line} on the curved surface,
forms a line which will correspond to the line on the curved surface. And, likewise,
to every finite figure on the curved surface will correspond a finite figure on the
auxiliary sphere, the area of which upon the latter shall be regarded as the measure
of the amplitude of the former. We shall either regard this area as a number, in
which case the square of the radius of the auxiliary sphere is the unit, or else
express it in degrees,~etc., setting the area of the hemisphere equal to~$360°$.

The comparison of the area upon the curved surface with the corresponding
amplitude leads to the idea of what we call the measure of curvature of the surface.
If the former is proportional to the latter, the curvature is called uniform;
and the quotient, when we divide the amplitude by the surface, is called the measure
of curvature. This is the case when the curved surface is a sphere, and the measure
of curvature is then a fraction whose numerator is unity and whose denominator is
the square of the radius.

We shall regard the measure of curvature as positive, if the boundaries of the
figures upon the curved surface and upon the auxiliary sphere run in the same sense;
as negative, if the boundaries enclose the figures in contrary senses. If they are not
proportional, the surface is \Typo{non-uniformily}{non-uniformly} curved. And at each point there exists a
particular measure of curvature, which is obtained from the comparison of corresponding
infinitesimal parts upon the curved surface and the auxiliary sphere. Let $d\sigma$~be
a surface element on the former, and $d\Sigma$~the corresponding element upon the auxiliary
sphere, then
\[
\frac{d\Sigma}{d\sigma}
\]
will be the measure of curvature at this point.

In order to determine their boundaries, we first project both upon the $xy$-plane.
The magnitudes of these projections are $Z\, d\sigma$,~$Z\, d\Sigma$. The sign of~$Z$ will show whether
the boundaries run in the same sense or in contrary senses around the surfaces and
their projections. We will suppose that the figure is a triangle; the projection upon
the $xy$-plane has the coordinates
\[
x,\ y;\qquad
x + dx,\ y + dy;\qquad
x + \delta x,\ y + \delta y.
\]
Hence its double area will be
\[
2Z\, d\sigma = dx·\delta y - dy·\delta x.
\]
To the projection of the corresponding element upon the sphere will correspond the
coordinates:
\PageSep{96}
\[
\begin{gathered}
X, \\
X + \frac{\dd X}{\dd x}·dx + \frac{\dd X}{\dd y}·dy, \\
X + \frac{\dd X}{\dd x}·\delta x + \frac{\dd X}{\dd y}·\delta y,
\end{gathered}
\qquad
\begin{gathered}
Y, \\
Y + \frac{\dd Y}{\dd x}·dx + \frac{\dd Y}{\dd y}·dy, \\
Y + \frac{\dd Y}{\dd x}·\delta x + \frac{\dd Y}{\dd y}·\delta y,
\end{gathered}
\]
From this the double area of the element is found to be
\begin{align*}
2Z\, d\Sigma
  &= \Neg
  \left(\frac{\dd X}{\dd x}·dx
      + \frac{\dd X}{\dd y}·dy\right)
  \left(\frac{\dd Y}{\dd x}·\delta x
      + \frac{\dd Y}{\dd y}·\delta y\right) \\
  &\phantom{={}}
 -\left(\frac{\dd X}{\dd x}·\delta x
      + \frac{\dd X}{\dd y}·\delta y\right)
  \left(\frac{\dd Y}{\dd x}·dx
      + \frac{\dd Y}{\dd y}·dy\right) \\
  &= \Neg
  \left(\frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
      - \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x}\right)
    (dx·\delta y - dy·\delta x).
\end{align*}
The measure of curvature is, therefore,
\[
= \frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
- \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x} = \omega.
\]
Since
\begin{gather*}
X = -tZ,\qquad
Y = -uZ, \\
(1 + t^{2} + u^{2})Z^{2} = 1,
\end{gather*}
we have
\begin{align*}
dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3}tu·du, \\
dY &= +Z^{3}tu·dt - Z^{3}(1 + t^{2})\, du,
\end{align*}
therefore
\begin{alignat*}{2}
\frac{\dd X}{\dd x}
  &= Z^{3}\bigl\{-(1 + u^{2})T + tuU\bigr\},\qquad&
\frac{\dd Y}{\dd x}
  &= Z^{3}\bigl\{tuT - (1 + t^{2})U\bigr\}, \\
%
\frac{\dd X}{\dd y}
  &= Z^{3}\bigl\{-(1 + u^{2})U + tuV\bigr\},\qquad&
\frac{\dd Y}{\dd y}
  &= Z^{3}\bigl\{tuU - (1 + t^{2})V\bigr\},
\end{alignat*}
and
\begin{align*}
\omega
  &= Z^{6}(TV - U^{2})\bigl((1 + t^{2})(1 + u^{2}) - t^{2}u^{2}\bigr) \\
  &= Z^{6}(TV - U^{2})(1 + t^{2} + u^{2}) \\
  &= Z^{4}(TV - U^{2}) \\
  &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}},
\end{align*}
the very same expression which we have found at the end of the preceding article.
Therefore we see that
\PageSep{97}

%[** TN: Quoted, not italicized, in the original]
\begin{Theorem}[]
The measure of curvature is always expressed by means of a fraction whose
numerator is unity and whose denominator is the product of the maximum
and minimum radii of curvature in the planes passing through the normal.
\end{Theorem}


\Article{12.}

We will now investigate the nature of shortest lines upon curved surfaces. The
nature of a curved line in space is determined, in general, in such a way that the
coordinates $x$,~$y$,~$z$ of each point are regarded as functions of a single variable, which
we shall call~$w$. The length of the curve, measured from an arbitrary origin to this
point, is then equal to
\[
\int \SQRT{\left(\frac{dx}{dw}\right)^{2}
  + \left(\frac{dy}{dw}\right)^{2}
  + \left(\frac{dz}{dw}\right)^{2}}·dw.
\]
If we allow the curve to change its position by an infinitely small variation, the variation
of the whole length will then be
{\small
\begin{multline*}
= \int \frac{\dfrac{dx}{dw}·d\, \delta x
           + \dfrac{dy}{dw}·d\, \delta y
           + \dfrac{dz}{dw}·d\, \delta z}
            {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
                 + \left(\dfrac{dy}{dw}\right)^{2}
                 + \left(\dfrac{dz}{dw}\right)^{2}}}% \\
%
= \frac{\dfrac{dx}{dw}·\delta x
      + \dfrac{dy}{dw}·\delta y
      + \dfrac{dz}{dw}·\delta z}
       {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
            + \left(\dfrac{dy}{dw}\right)^{2}
            + \left(\dfrac{dz}{dw}\right)^{2}}} \displaybreak[1] \\
\qquad- \int\left\{
  \delta x·d\frac{\dfrac{dx}{dw}}
    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
         + \left(\dfrac{dy}{dw}\right)^{2}
         + \left(\dfrac{dz}{dw}\right)^{2}}}\right.
+ \delta y·d\frac{\dfrac{dy}{dw}}
    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
         + \left(\dfrac{dy}{dw}\right)^{2}
         + \left(\dfrac{dz}{dw}\right)^{2}}} \\
\qquad\qquad\qquad+ \left.\delta z·d\frac{\dfrac{dz}{dw}}
    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
         + \left(\dfrac{dy}{dw}\right)^{2}
         + \left(\dfrac{dz}{dw}\right)^{2}}}\right\}.
\end{multline*}}%
The expression under the integral sign must vanish in the case of a minimum, as we
know. Since the curved line lies upon a given curved surface whose equation is
\[
P\, dx + Q\, dy + R\, dz = 0,
\]
the equation between the variations $\delta x$,~$\delta y$,~$\delta z$
\[
P\, \delta z + Q\, \delta y + R\, \delta z = 0
\]
must also hold. From this, by means of well known principles, we easily conclude
that the differentials
\PageSep{98}
\begin{gather*}
  d·\frac{\dfrac{dx}{dw}}
           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
                + \left(\dfrac{dy}{dw}\right)^{2}
                + \left(\dfrac{dz}{dw}\right)^{2}}}, \\
  d·\frac{\dfrac{dy}{dw}}
           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
                + \left(\dfrac{dy}{dw}\right)^{2}
                + \left(\dfrac{dz}{dw}\right)^{2}}}, \\
  d·\frac{\dfrac{dz}{dw}}
           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
                + \left(\dfrac{dy}{dw}\right)^{2}
                + \left(\dfrac{dz}{dw}\right)^{2}}}
\end{gather*}
must be proportional to the quantities $P$,~$Q$,~$R$ respectively. If $ds$~is an element of
the curve; $\lambda$~the point upon the auxiliary sphere, which represents the direction of
this element; $L$~the point giving the direction of the normal as above; and $\xi$,~$\eta$,~$\zeta$;
$X$,~$Y$,~$Z$ the coordinates of the points $\lambda$,~$L$ referred to the centre of the auxiliary
sphere, then we have
\begin{gather*}
dx = \xi\, ds,\qquad
dy = \eta\, ds,\qquad
dz = \zeta\, ds, \\
\xi^{2} + \eta^{2} + \zeta^{2} = 1.
\end{gather*}
Therefore we see that the above differentials will be equal to $d\xi$,~$d\eta$,~$d\zeta$. And since
$P$,~$Q$,~$R$ are proportional to the quantities $X$,~$Y$,~$Z$, the character of the shortest line
is such that
\[
\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}.
\]


\Article{13.}

To every point of a curved line upon a curved surface there correspond two
points on the sphere, according to our point of view; namely, the point~$\lambda$, which
represents the direction of the linear element, and the point~$L$, which represents the
direction of the normal to the surface. The two are evidently $90°$~apart. In our
former investigation (\Art{9}), where [we] supposed the curved line to lie in a plane,
we had \emph{two} other points upon the sphere; namely,~$\L$, which represents the direction
of the normal to the plane, and~$\lambda'$, which represents the direction of the normal to
the element of the curve in the plane. In this case, therefore, $\L$~was a fixed point
and $\lambda$,~$\lambda'$ were always in a great circle whose pole was~$\L$. In generalizing these
considerations, we shall retain the notation $\L$,~$\lambda'$, but we must define the meaning of
these symbols from a more general point of view. When the curve~$s$ is described,
the points $L$,~$\lambda$ also describe curved lines upon the auxiliary sphere, which, generally
speaking, are no longer great circles. Parallel to the element of the second line,
\PageSep{99}
we draw a radius of the auxiliary sphere to the point~$\lambda'$, but instead of this point
we take the point opposite when $\lambda'$~is more than~$90°$ from~$L$. In the first case, we
regard the element at~$\lambda$ as positive, and in the other as negative. Finally, let $\L$ be
the point on the auxiliary sphere, which is $90°$~from both $\lambda$~and~$\lambda'$, and which is so
taken that $\lambda$,~$\lambda'$,~$\L$ lie in the same order as $(1)$,~$(2)$,~$(3)$.

The coordinates of the four points of the auxiliary sphere, referred to its centre,
are for
\begin{alignat*}{4}
&L\qquad &&X\quad &&Y\quad &&Z \\
&\lambda &&\xi    &&\eta   &&\zeta \\
&\lambda'&&\xi'   &&\eta'  &&\zeta' \\
&\L      &&\alpha &&\beta  &&\gamma.
\end{alignat*}
Hence each of these $4$~points describes a line upon the auxiliary sphere, whose elements
we shall express by $dL$,~$d\lambda$,~$d\lambda'$,~$d\L$. We have, therefore,
\begin{align*}
d\xi   &= \xi'\, d\lambda, \\
d\eta  &= \eta'\, d\lambda, \\
d\zeta &= \zeta'\, d\lambda.
\end{align*}
In an analogous way we now call
\[
\frac{d\lambda}{ds}
\]
the measure of curvature of the curved line upon the curved surface, and its reciprocal
\[
\frac{ds}{d\lambda}
\]
the radius of curvature. If we denote the latter by~$\rho$, then
\begin{align*}
\rho\, d\xi   &= \xi'\, ds, \\
\rho\, d\eta  &= \eta'\, ds, \\
\rho\, d\zeta &= \zeta'\, ds.
\end{align*}

If, therefore, our line be a shortest line, $\xi'$,~$\eta'$,~$\zeta'$ must be proportional to the
quantities $X$,~$Y$,~$Z$. But, since at the same time
\[
\xi'^{2} + \eta'^{2} + \zeta'^{2}  = X^{2} + Y^{2} + Z^{2} = 1,
\]
we have
\[
\xi'   = ±X,\quad
\eta'  = ±Y,\quad
\zeta' = ±Z,
\]
and since, further,
\begin{align*}
\xi'X + \eta'Y + \zeta'Z
  &= \cos \lambda'L \\
  &= ±(X^{2} + Y^{2} + Z^{2}) \\
  &= ±1,
\end{align*}
\PageSep{100}
and since we always choose the point~$\lambda'$ so that
\[
\lambda'L < 90°,
\]
then for the shortest line
\[
\lambda'L = 0,
\]
or $\lambda'$~and~$L$ must coincide. Therefore
\begin{align*}
\rho\, d\xi   &= X\, ds, \\
\rho\, d\eta  &= Y\, ds, \\
\rho\, d\zeta &= Z\, ds,
\end{align*}
and we have here, instead of $4$~curved lines upon the auxiliary sphere, only $3$~to consider.
Every element of the second line is therefore to be regarded as lying in the
great circle~$L\lambda$. And the positive or negative value of~$\rho$ refers to the concavity
or the convexity of the curve in the direction of the normal.


\Article{14.}

We shall now investigate the spherical angle upon the auxiliary sphere, which
the great circle going from~$L$ toward~$\lambda$ makes with that one going from~$L$ toward
one of the fixed points $(1)$,~$(2)$,~$(3)$; \eg, toward~$(3)$. In order to have something
definite here, we shall consider the sense from~$L(3)$ to~$L\lambda$ the same as that in which
$(1)$,~$(2)$, and~$(3)$ lie. If we call this angle~$\phi$, then it follows from the theorem of \Art{7}
that\Note{37}
\[
\sin L(3)·\sin L\lambda·\sin\phi = Y\xi - X\eta,\NoteMark
\]
or, since $L\lambda = 90°$ and
\[
\sin L(3) = \Sqrt{X^{2} + Y^{2}} = \Sqrt{1 - Z^{2}},
\]
we have
\[
\sin\phi = \frac{Y\xi - X\eta}{\Sqrt{X^{2} + Y^{2}}}.
\]
Furthermore,
\[
\sin L(3)·\sin L\lambda·\cos\phi = \zeta,
\]
or
\[
\cos\phi = \frac{\zeta}{\Sqrt{X^{2} + Y^{2}}}
\]
and
\[
\tan\phi = \frac{Y\xi - X\eta}{\zeta} = \frac{\zeta'}{\zeta}.
\]
\PageSep{101}
Hence we have
\[
d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta
  - (Y\xi - X\eta)\, d\zeta + \xi\zeta\, dY - \eta\zeta\, dX}
  {(Y\xi - X\eta)^{2} + \zeta^{2}}.
\]
The denominator of this expression is
\begin{align*}
&= Y^{2}\xi^{2} - 2XY\xi\eta - X^{2}\eta^{2} + \zeta^{2} \\
&= -(X\xi + Y\eta)^{2} + (X^{2} + Y^{2})(\xi^{2} + \eta^{2}) + \zeta^{2} \\
&= -Z^{2}\zeta^{2} + (1 - Z^{2})(1 - \zeta^{2}) + \zeta^{2} \\
&= 1 - Z^{2},
\end{align*}
or
\[
d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta
  + (X\eta - Y\xi)\, d\zeta - \eta\zeta\, dX + \xi\zeta\, dY}
  {1 - Z^{2}}.
\]

We verify readily by expansion the identical equation
\begin{gather*}
\eta\zeta(X^{2} + Y^{2} + Z^{2}) + YZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\
= (X\xi + Y\eta + Z\zeta)(Z\eta + Y\zeta) + (X\zeta - Z\xi)(X\eta - Y\xi)\Add{,}
\end{gather*}
and likewise
\begin{gather*}
\xi\zeta(X^{2} + Y^{2} + Z^{2}) + XZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\
= (X\xi + Y\eta + Z\zeta)(X\zeta + Z\xi) + (Y\xi - X\eta)(Y\zeta - Z\eta).
\end{gather*}
We have, therefore,
\begin{align*}
\eta\zeta &= -YZ + (X\zeta - Z\xi )(X\eta  - Y\xi), \\
\xi\zeta  &= -XZ + (Y\xi   - X\eta)(Y\zeta - Z\eta).
\end{align*}
Substituting these values, we obtain
\begin{multline*}
d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)
  + \frac{\zeta Y\, d\xi - \zeta X\, d\eta}{1 - Z^{2}} \\
  + \frac{X\eta - Y\xi}{1 - Z^{2}}\bigl\{
    d\zeta - (X\zeta - Z\xi)\, dX - (Y\zeta - Z\eta)\, dY\bigr\}.
\end{multline*}
Now
\begin{alignat*}{4}
&  X\, dX &&+    Y\, dY &&+     Z\, dZ &&= 0, \\
&\xi\, dX &&+ \eta\, dY &&+ \zeta\, dZ &&= -X\, d\xi - Y\, d\eta - Z\, d\zeta.
\end{alignat*}
On substituting we obtain, instead of what stands in the parenthesis,
\[
d\zeta - Z(X\, d\xi + Y\, d\eta + Z\, d\zeta).
\]
Hence\Note{38}
\begin{align*}
d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)
  &+ \frac{d\xi}{1 - Z^{2}}\{\zeta Y - \eta X^{2}Z + \xi XYZ\} \\
  &- \frac{d\eta}{1 - Z^{2}}\{\zeta X + \eta XYZ - \xi Y^{2}Z\}\NoteMark \\
  &+ d\zeta(\eta X - \xi Y).
\end{align*}
\PageSep{102}
Since, further,
\begin{align*}
\eta X^{2}Z - \xi XYZ
  &= \eta X^{2}Z + \eta Y^{2}Z + \zeta ZYZ \\
  &= \eta Z(1 - Z^{2}) + \zeta YZ^{2}, \\
%
\eta XYZ - \xi Y^{2}Z
  &= -\xi X^{2}Z - \zeta XZ^{2} - \xi Y^{2}Z \\
  &= - \xi Z(1 - Z^{2}) - \zeta XZ^{2},
\end{align*}
our whole expression becomes
\begin{align*}
d\phi &= \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY) \\
&\quad
  + (\zeta Y - \eta  Z)\, d\xi
  + (\xi   Z - \zeta X)\, d\eta
  + (\eta  X - \xi   Y)\, d\zeta.
\end{align*}


\Article{15.}

The formula just found is true in general, whatever be the nature of the curve.
But if this be a shortest line, then it is clear that the last three terms destroy each
other, and consequently\Note{39}
\[
d\phi = -\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX).\NoteMark
\]
But we see at once that
\[
\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX)
\]
is nothing but the area of the part of the auxiliary sphere, which is formed between
the element of the line~$L$, the two great circles drawn through its extremities and~$(3)$,\Note{40}
%[Illustration]
\Figure{102}
and the element thus intercepted on the great circle through $(1)$~and~$(2)$. This
surface is considered positive, if $L$~and~$(3)$ lie on the same side of~$(1)\ (2)$, and if the
\PageSep{103}
direction from~$P$ to~$P'$ is the same as that from~$(2)$ to~$(1)$; negative, if the contrary
of one of these conditions hold; positive again, if the contrary of both conditions be
true. In other words, the surface is considered positive if we go around the circumference
of the figure~$LL'P'P$ in the same sense as $(1)\ (2)\ (3)$; negative, if we go
in the contrary sense.

If\Note{41} we consider now a finite part of the line from~$L$ to~$L'$ and denote by $\phi$,~$\phi'$
the values of the angles at the two extremities, then we have
\[
\phi' = \phi + \Area LL'P'P,
\]
the sign of the area being taken as explained.

Now\Note{42} let us assume further that, from the origin upon the curved surface, infinitely
many other shortest lines go out, and denote by~$A$ that indefinite angle which the
first element, moving counter-clockwise, makes with the first element of the first line;
and through the other extremities of the different curved lines let a curved line be drawn,
concerning which, first of all, we leave it undecided whether it be a shortest line or
not. If we suppose also that those indefinite values, which
for the first line were $\phi$,~$\phi'$, be denoted by $\psi$,~$\psi'$ for each of
these lines, then $\psi' - \psi$ is capable of being represented in
the same manner on the auxiliary sphere by the space~$LL'_{1}P'_{1}P$.
Since evidently $\psi = \phi - A$, the space\Note{43}
\[
\begin{aligned}[b]
LL'_{1}P'_{1}P'L'L
  &= \psi' - \psi - \phi' + \phi \\
  &= \psi' - \phi' + A \\
  &= LL'_{1}L'L + L'L'_{1}P'_{1}P'.\NoteMark
\end{aligned}
\qquad\qquad
%[Illustration]
\raisebox{-\baselineskip}{\Graphic{1.5in}{103}}
\]

If the bounding line is also a shortest line, and, when prolonged, makes with
$LL'$,~$LL'_{1}$ the angles $B$,~$B_{1}$; if, further, $\chi$,~$\chi_{1}$ denote the same at the points $L'$,~$L'_{1}$,
that $\phi$~did at~$L$ in the line~$LL'$, then we have
\begin{align*}
\chi_{1} &= \chi + \Area L'L'_{1}P'_{1}P', \\
\psi' - \phi' + A &= LL'_{1}L'L + \chi_{1} - \chi;
\end{align*}
but
\begin{align*}
\phi' &= \chi + B, \\
\psi' &= \chi_{1} + B_{1},
\end{align*}
therefore
\[
B_{1} - B + A = LL'_{1}L'L.
\]
The angles of the triangle~$LL'L'_{1}$ evidently are
\[
A,\qquad 180° - B,\qquad B_{1},
\]
\PageSep{104}
therefore their sum is
\[
180° + LL'_{1}L'L.
\]

The form of the proof will require some modification and explanation, if the point~$(3)$
falls within the triangle. But, in general, we conclude

%[** TN: Quoted, not italicized, in the original]
\begin{Theorem}[]
The sum of the three angles of a triangle, which is formed of shortest lines
upon an arbitrary curved surface, is equal to the sum of~$180°$ and the area of
the triangle upon the auxiliary sphere, the boundary of which is formed by the
points~$L$, corresponding to the points in the boundary of the original triangle,
and in such a manner that the area of the triangle may be regarded as positive
or negative according as it is inclosed by its boundary in the same sense as
the original figure or the contrary.
\end{Theorem}

Wherefore\Note{44} we easily conclude also that the sum of all the angles of a polygon
of $n$~sides, which are shortest lines upon the curved surface, is [equal to] the sum
of $(n - 2)180° + \text{the area of the polygon upon the sphere~etc.}$


\Article{16.}

If one curved surface can be completely developed upon another surface, then all
lines upon the first surface will evidently retain their magnitudes after the development
upon the other surface; likewise the angles which are formed by the intersection
of two lines. Evidently, therefore, such lines also as are shortest lines upon
one surface remain shortest lines after the development. Whence, if to any arbitrary
polygon formed of shortest lines, while it is upon the first surface, there corresponds
the figure of the zeniths\Note{45} upon the auxiliary sphere, the area of which is~$A$,
and if, on the other hand, there corresponds to the same polygon, after its development
upon another surface, a figure of the zeniths upon the auxiliary sphere, the
area of which is~$A'$, it follows at once that in every case
\[
A = A'.
\]
Although this proof originally presupposes the boundaries of the figures to be shortest
lines, still it is easily seen that it holds generally, whatever the boundary may be.
For, in fact, if the theorem is independent of the number of sides, nothing will prevent
us from imagining for every polygon, of which some or all of its sides are not
shortest lines, another of infinitely many sides all of which are shortest lines.

Further, it is clear that every figure retains also its area after the transformation
by development.
\PageSep{105}

We shall here consider 4~figures:

%[** TN: Indented line list items in the original]
\Par{1)} an arbitrary figure upon the first surface,

\Par{2)} the figure on the auxiliary sphere, which corresponds to the zeniths of the
previous figure,

\Par{3)} the figure upon the second surface, which No.~1 forms by the development,

\Par{4)} the figure upon the auxiliary sphere, which corresponds to the zeniths of
No.~3.

Therefore, according to what we have proved, 2~and~4 have equal areas, as also
1~and~3. Since we assume these figures infinitely small, the quotient obtained by
dividing 2~by~1 is the measure of curvature of the first curved surface at this point,
and likewise the quotient obtained by dividing 4~by~3, that of the second surface.
From this follows the important theorem:

%[** TN: Quoted, not italicized in the original]
\begin{Theorem}[]
In the transformation of surfaces by development the measure of curvature
at every point remains unchanged.
\end{Theorem}
This is true, therefore, of the product of the greatest and smallest radii of curvature.

In the case of the plane, the measure of curvature is evidently everywhere zero.
Whence follows therefore the important theorem:

\begin{Theorem}[]
For all surfaces developable upon a plane the measure of curvature everywhere
vanishes,
\end{Theorem}
or
\[
\left(\frac{\dd^{2}z}{\dd x\, \dd y}\right)^{2}
  - \left(\frac{\dd^{2} z}{\dd x^{2}}\right)
    \left(\frac{\dd^{2} z}{\dd x^{2}}\right) = 0,
\]
which criterion is elsewhere derived from other principles, though, as it seems to us,
not with the desired rigor. It is clear that in all such surfaces the zeniths of all
points can not fill out any space, and therefore they must all lie in a line.


\Article{17.}

From a given point on a curved surface we shall let an infinite number of shortest
lines go out, which shall be distinguished from one another by the angle which their
first elements make with the first element of a \emph{definite} shortest line. This angle we
shall call~$\theta$. Further, let $s$~be the length [measured from the given point] of a part
of such a shortest line, and let its extremity have the coordinates $x$,~$y$,~$z$. Since $\theta$~and~$s$,
therefore, belong to a perfectly definite point on the curved surface, we can
regard $x$,~$y$,~$z$ as functions of $\theta$~and~$s$. The direction of the element of~$s$ corresponds
to the point~$\lambda$ on the sphere, whose coordinates are $\xi$,~$\eta$,~$\zeta$. Thus we shall have
\PageSep{106}
\[
\xi = \frac{\dd x}{\dd s},\qquad
\eta = \frac{\dd y}{\dd s},\qquad
\zeta = \frac{\dd z}{\dd s}.
\]

The extremities of all shortest lines of equal lengths~$s$ correspond to a curved
line whose length we may call~$t$. We can evidently consider~$t$ as a function of $s$~and~$\theta$,
and if the direction of the element of~$t$ corresponds upon the sphere to the point~$\lambda'$
whose coordinates are $\xi'$,~$\eta'$,~$\zeta'$, we shall have
\[
\xi'·\frac{\dd t}{\dd\theta} = \frac{\dd x}{\dd\theta},\qquad
\eta'·\frac{\dd t}{\dd\theta} = \frac{\dd y}{\dd\theta},\qquad
\zeta'·\frac{\dd t}{\dd\theta} = \frac{\dd z}{\dd\theta}.
\]
Consequently
\[
(\xi\xi' + \eta\eta' + \zeta\zeta')\, \frac{\dd t}{\dd\theta}
  = \frac{\dd x}{\dd s}·\frac{\dd x}{\dd\theta}
  + \frac{\dd y}{\dd s}·\frac{\dd y}{\dd\theta}
  + \frac{\dd z}{\dd s}·\frac{\dd z}{\dd\theta}.
\]
This magnitude we shall denote by~$u$, which itself, therefore, will be a function of $\theta$~and~$s$.

We find, then, if we differentiate with respect to~$s$,
\begin{align*}
\frac{\dd u}{\dd s}
  &= \frac{\dd^{2} x}{\dd s^{2}}·\frac{\dd x}{\dd\theta}
   + \frac{\dd^{2} y}{\dd s^{2}}·\frac{\dd y}{\dd\theta}
   + \frac{\dd^{2} z}{\dd s^{2}}·\frac{\dd z}{\dd\theta}
   + \tfrac{1}{2}\, \frac{\dd\left\{
       \left(\dfrac{\dd x}{\dd s}\right)^{2}
     + \left(\dfrac{\dd y}{\dd s}\right)^{2}
     + \left(\dfrac{\dd z}{\dd s}\right)^{2}\right\}}{\dd\theta} \\
  &= \frac{\dd^{2} x}{\dd s^{2}}·\frac{\dd x}{\dd\theta}
   + \frac{\dd^{2} y}{\dd s^{2}}·\frac{\dd y}{\dd\theta}
   + \frac{\dd^{2} z}{\dd s^{2}}·\frac{\dd z}{\dd\theta},
\end{align*}
because
\[
  \left(\dfrac{\dd x}{\dd s}\right)^{2}
+ \left(\dfrac{\dd y}{\dd s}\right)^{2}
+ \left(\dfrac{\dd z}{\dd s}\right)^{2} = 1,
\]
and therefore its differential is equal to zero.

But since all points [belonging] to one constant value of~$\theta$ lie on a shortest line,
if we denote by~$L$ the zenith of the point to which $s$,~$\theta$ correspond and by $X$,~$Y$,~$Z$
the coordinates of~$L$, [from the last formulæ of \Art{13}],
\[
\frac{\dd^{2} x}{\dd s^{2}} = \frac{X}{p},\qquad
\frac{\dd^{2} y}{\dd s^{2}} = \frac{Y}{p},\qquad
\frac{\dd^{2} z}{\dd s^{2}} = \frac{Z}{p},
\]
if $p$~is the radius of curvature. We have, therefore,
\[
p·\frac{\dd u}{\dd s}
  = X\, \frac{\dd x}{\dd\theta}
  + Y\, \frac{\dd y}{\dd\theta}
  + Z\, \frac{\dd z}{\dd\theta}
  = \frac{\dd t}{\dd\theta}(X\xi' + Y\eta' + Z\zeta').
\]
But
\[
X\xi' + Y\eta' + Z\zeta' = \cos L\lambda' = 0,
\]
because, evidently, $\lambda'$~lies on the great circle whose pole is~$L$. Therefore we have
\[
\frac{\dd u}{\dd s} = 0,
\]
\PageSep{107}
or $u$~independent of~$s$, and therefore a function of $\theta$~alone. But for $s = 0$, it is evident
that $t = 0$, $\dfrac{\dd t}{\dd\theta} = 0$, and therefore $u = 0$. Whence we conclude that, in general,
$u = 0$, or
\[
\cos \lambda\lambda' = 0.
\]
From this follows the beautiful theorem:

\begin{Theorem}[]
If all lines drawn from a point on the curved surface are shortest lines of
equal lengths, they meet the line which joins their extremities everywhere at
right angles.
\end{Theorem}

We can show in a similar manner that, if upon the curved surface any curved
line whatever is given, and if we suppose drawn from every point of this line toward
the same side of it and at right angles to it only shortest lines of equal lengths, the
extremities of which are joined by a line, this line will be cut at right angles by
those lines in all its points. We need only let $\theta$ in the above development represent
the length of the \emph{given} curved line from an arbitrary point, and then the above calculations
retain their validity, except that $u = 0$ for $s = 0$ is now contained in the
hypothesis.


\Article{18.}

The relations arising from these constructions deserve to be developed still more
fully. We have, in the first place, if, for brevity, we write~$m$ for~$\dfrac{\dd t}{\dd\theta}$,
\begin{alignat*}{3}
\Tag{(1)}
\frac{\dd x}{\dd s} &= \xi, &
\frac{\dd y}{\dd s} &= \eta, &
\frac{\dd z}{\dd s} &= \zeta, \\
\Tag{(2)}
\frac{\dd x}{\dd\theta} &= m\xi',\quad &
\frac{\dd y}{\dd\theta} &= m\eta',\quad &
\frac{\dd z}{\dd\theta} &= m\zeta',
\end{alignat*}
\begin{alignat*}{4}
\Tag{(3)}
&\xi^{2}  &&+ \eta^{2}  &&+ \zeta^{2} &&= 1, \\
\Tag{(4)}
&\xi'^{2} &&+ \eta'^{2} &&+ \zeta'^{2} &&= 1, \\
\Tag{(5)}
&\xi\xi'  &&+ \eta\eta' &&+ \zeta\zeta' &&= 0.
\end{alignat*}
Furthermore,
\begin{alignat*}{4}
\Tag{(6)}
&X^{2} &&+ Y^{2} &&+ Z^{2} &&= 1, \\
\Tag{(7)}
&X\xi  &&+ Y\eta &&+ Z\zeta &&= 0, \\
\Tag{(8)}
&X\xi' &&+ Y\eta' &&+ Z\zeta' &&= 0,
\end{alignat*}
and
\begin{align*}
\Tag{[9]}
&\left\{
\begin{alignedat}{2}
X &= \zeta\eta' &&- \eta\zeta', \\
Y &= \xi\zeta'  &&- \zeta\xi', \\
Z &= \eta\xi'   &&- \xi\eta';
\end{alignedat}
\right. \\
\PageSep{108}
\Tag{[10]}
&\left\{
\begin{alignedat}{2}
\xi'   &= \eta  Z &&- \zeta Y, \\
\eta'  &= \zeta X &&- \xi   Z, \\
\zeta' &= \xi   Y &&- \eta  X;
\end{alignedat}
\right. \\
\Tag{[11]}
&\left\{
\begin{alignedat}{2}
\xi   &= Y\zeta' &&- Z\eta', \\
\eta  &= Z\xi'   &&- X\zeta', \\
\zeta &= X\eta'  &&- Y\xi'.
\end{alignedat}
\right.
\end{align*}

Likewise, $\dfrac{\dd\xi}{\dd s}$, $\dfrac{\dd\eta}{\dd s}$, $\dfrac{\dd\zeta}{\dd s}$ are proportional to $X$,~$Y$,~$Z$, and if we set
\[
\frac{\dd\xi}{\dd s}   = pX,\qquad
\frac{\dd\eta}{\dd s}  = pY,\qquad
\frac{\dd\zeta}{\dd s} = pZ,
\]
where $\dfrac{1}{p}$ denotes the radius of curvature of the line~$s$, then
\[
p = X\, \frac{\dd\xi}{\dd s}
  + Y\, \frac{\dd\eta}{\dd s}
  + Z\, \frac{\dd\zeta}{\dd s}.
\]
By differentiating~(7) with respect to~$s$, we obtain
\[
-p = \xi\, \frac{\dd X}{\dd s}
   + \eta\, \frac{\dd Y}{\dd s}
   + \zeta\, \frac{\dd Z}{\dd s}.
\]

We can easily show that $\dfrac{\dd\xi'}{\dd s}$, $\dfrac{\dd\eta'}{\dd s}$, $\dfrac{\dd\zeta'}{\dd s}$ also are proportional to $X$,~$Y$,~$Z$. In fact,
[from~10] the values of these quantities are also [equal to]
\[
\eta\, \frac{\dd Z}{\dd s} - \zeta\, \frac{\dd Y}{\dd s},\qquad
\zeta\, \frac{\dd X}{\dd s} - \xi\, \frac{\dd Z}{\dd s},\qquad
\xi\, \frac{\dd Y}{\dd s} - \eta\, \frac{\dd X}{\dd s},
\]
therefore
\begin{align*}
Y\, \frac{\dd\xi'}{\dd s} - X\, \frac{\dd\eta'}{\dd s}
  &= - \zeta\left(\frac{Y\, \dd Y}{\dd s} + \frac{X\, \dd X}{\dd s}\right)
   + \frac{\dd Z}{\dd s}(Y\eta + X\xi) \\
  &= - \zeta\left(\frac{X\, \dd X + Y\, \dd Y + Z\, \dd Z}{\dd s}\right)
   + \frac{\dd Z}{\dd s}(X\xi + Y\eta + Z\zeta) \\
  &= 0,
\end{align*}
and likewise the others. We set, therefore,
\[
\frac{\dd\xi'}{\dd s} = p'X,\qquad
\frac{\dd\eta'}{\dd s} = p'Y,\qquad
\frac{\dd\zeta'}{\dd s} = p'Z,
\]
whence
\[
p' = ±\SQRT{\left(\frac{\dd\xi'}{\dd s}\right)^{2}
  + \left(\frac{\dd\eta'}{\dd s}\right)^{2}
  + \left(\frac{\dd\zeta'}{\dd s}\right)^{2}}\Add{,}
\]
\PageSep{109}
and also
\[
p' = X\, \frac{\dd\xi'}{\dd s}
   + Y\, \frac{\dd\eta'}{\dd s}
   + Z\, \frac{\dd\zeta'}{\dd s}.
\]
Further [we obtain], from the result obtained by differentiating~(8),
\[
-p' = \xi'\, \frac{\dd X}{\dd s}
    + \eta'\, \frac{\dd Y}{\dd s}
    + \zeta'\, \frac{\dd Z}{\dd s}.
\]
But we can derive two other expressions for this. We have
\[
\frac{\dd m\xi'}{\dd s} = \frac{\dd\xi}{\dd\theta},\qquad
\left[
\frac{\dd m\eta'}{\dd s} = \frac{\dd\eta}{\dd\theta},\qquad
\frac{\dd m\zeta'}{\dd s} = \frac{\dd\zeta}{\dd\theta},
\right]
\]
therefore [because of~(8)]
\[
mp' = X\, \frac{\dd\xi}{\dd\theta}
    + Y\, \frac{\dd\eta}{\dd\theta}
    + Z\, \frac{\dd\zeta}{\dd\theta}.
\]
[and therefore, from~(7),]
\[
-mp' = \xi\, \frac{\dd X}{\dd\theta}
     + \eta\, \frac{\dd Y}{\dd\theta}
     + \zeta\, \frac{\dd Z}{\dd\theta}.
\]

After these preliminaries [using (2)~and~(4)] we shall now first put~$m$ in the form
\[
m = \xi'\, \frac{\dd x}{\dd\theta}
  + \eta'\, \frac{\dd y}{\dd\theta}
  + \zeta'\, \frac{\dd z}{\dd\theta},
\]
and differentiating with respect to~$s$, we have\footnote
  {It is better to differentiate~$m^{2}$. [In fact from (2)~and~(4)
  \[
  m^{2} = \left(\frac{\dd x}{\dd\theta}\right)^{2}
        + \left(\frac{\dd y}{\dd\theta}\right)^{2}
        + \left(\frac{\dd z}{\dd\theta}\right)^{2},
  \]
  therefore
  \begin{align*}
  m\, \frac{\dd m}{\dd s}
   &= \frac{\dd x}{\dd\theta}·\frac{\dd^{2} x}{\dd\theta\, \dd s}
    + \frac{\dd y}{\dd\theta}·\frac{\dd^{2} y}{\dd\theta\, \dd s}
    + \frac{\dd z}{\dd\theta}·\frac{\dd^{2} z}{\dd\theta\, \dd s} \\
   &= m\xi'\, \frac{\dd\xi}{\dd\theta}
    + m\eta'\, \frac{\dd\eta}{\dd\theta}
    + m\zeta'\, \frac{\dd\zeta}{\dd\theta}.]
  \end{align*}}
% [** End of footnote]
\begin{align*}
%[** TN: Re-broken]
\frac{\dd m}{\dd s}
  &= \frac{\dd x}{\dd\theta}·\frac{\dd\xi'}{\dd s}
   + \frac{\dd y}{\dd\theta}·\frac{\dd\eta'}{\dd s}
   + \frac{\dd z}{\dd\theta}·\frac{\dd\zeta'}{\dd s} %\\
%  &\quad
   + \xi'\, \frac{\dd^{2} x}{\dd s\, \dd\theta}
        + \eta'\, \frac{\dd^{2} y}{\dd s\, \dd\theta}
        + \zeta'\, \frac{\dd^{2} z}{\dd s\, \dd\theta} \displaybreak[1] \\
%
  &= mp'(\xi'X + \eta'Y + \zeta'Z) %\\
%  &\quad
   + \xi'\, \frac{\dd\xi}{\dd\theta}
        + \eta'\, \frac{\dd\eta}{\dd\theta}
        + \zeta'\, \frac{\dd\zeta}{\dd\theta} \displaybreak[1] \\
%
  &= \xi'\, \frac{\dd\xi}{\dd\theta}
   + \eta'\, \frac{\dd\eta}{\dd\theta}
   + \zeta'\, \frac{\dd\zeta}{\dd\theta}.
\end{align*}
\PageSep{110}

If we differentiate again with respect to~$s$, and notice that
\[
\frac{\dd^{2} \xi}{\dd s\, \dd\theta}
  = \frac{\dd(pX)}{\dd\theta},\quad\text{etc.},
\]
and that
\[
X\xi' + Y\eta' + Z\zeta' = 0,
\]
we have\Note{46}
{\small
\begin{align*}
\frac{\dd^{2} m}{\dd s^{2}}
  &= p\left(\xi'\, \frac{\dd X}{\dd\theta}
          + \eta'\, \frac{\dd Y}{\dd\theta}
          + \zeta'\, \frac{\dd Z}{\dd\theta}\right)
   + p'\left(X \frac{\dd\xi}{\dd\theta}
          + Y \frac{\dd \eta}{\dd\theta}
          + Z \frac{\dd \zeta}{\dd\theta}\right) \\
%
  &= p\left(\xi'\, \frac{\dd X}{\dd\theta}
          + \eta'\, \frac{\dd Y}{\dd\theta}
          + \zeta'\, \frac{\dd Z}{\dd\theta}\right) + mp'^{2} \\
%
  &= -\left(\xi\, \frac{\dd X}{\dd s}
          + \eta\, \frac{\dd Y}{\dd s}
          + \zeta\, \frac{\dd Z}{\dd s}\right)
      \left(\xi'\, \frac{\dd X}{\dd\theta}
          + \eta'\, \frac{\dd Y}{\dd\theta}
          + \zeta'\, \frac{\dd Z}{\dd\theta}\right) \\
  &\phantom{={}}
    + \left(\xi'\, \frac{\dd X}{\dd s}
          + \eta'\, \frac{\dd Y}{\dd s}
          + \zeta'\, \frac{\dd Z}{\dd s}\right)
      \left(\xi\, \frac{\dd X}{\dd\theta}
          + \eta\, \frac{\dd Y}{\dd\theta}
          + \zeta\, \frac{\dd Z}{\dd\theta}\right) \\
%
  &= \left(\frac{\dd Y}{\dd\theta}\, \frac{\dd Z}{\dd s}
         - \frac{\dd Y}{\dd s}\, \frac{\dd Z}{\dd\theta}\right)X
   + \left(\frac{\dd Z}{\dd\theta}\, \frac{\dd X}{\dd s}
         - \frac{\dd Z}{\dd s}\, \frac{\dd X}{\dd\theta}\right)Y
   + \left(\frac{\dd X}{\dd\theta}\, \frac{\dd Y}{\dd s}
         - \frac{\dd X}{\dd s}\, \frac{\dd Y}{\dd\theta}\right)Z.\NoteMark
\end{align*}}

[But if the surface element
\[
m\, ds\, d\theta
\]
belonging to the point $x$,~$y$,~$z$ be represented upon the auxiliary sphere of unit radius
by means of parallel normals, then there corresponds to it an area whose magnitude is
{\small
\[
\left\{
X\left(\frac{\dd Y}{\dd s}\, \frac{\dd Z}{\dd\theta}
     - \frac{\dd Y}{\dd\theta}\, \frac{\dd Z}{\dd s}\right) +
Y\left(\frac{\dd Z}{\dd s}\, \frac{\dd X}{\dd\theta}
     - \frac{\dd Z}{\dd\theta}\, \frac{\dd X}{\dd s}\right) +
Z\left(\frac{\dd X}{\dd s}\, \frac{\dd Y}{\dd\theta}
     - \frac{\dd X}{\dd\theta}\, \frac{\dd Y}{\dd s}\right)
\right\}ds\, d\theta.
\]}%
Consequently, the measure of curvature at the point under consideration is equal to
\[
-\frac{1}{m}\, \frac{\dd^{2} m}{\dd s^{2}}.]
\]
\PageSep{111}


\Notes.

The parts enclosed in brackets are additions of the editor of the German edition
or of the translators.

``The foregoing fragment, \textit{Neue allgemeine Untersuchungen über die krummen Flächen},
differs from the \textit{Disquisitiones} not only in the more limited scope of the matter, but
also in the method of treatment and the arrangement of the theorems. There [paper
of~1827] \textsc{Gauss} assumes that the rectangular coordinates $x$,~$y$,~$z$ of a point of the surface
can be expressed as functions of any two independent variables $p$~and~$q$, while
here [paper of~1825] he chooses as new variables the geodesic coordinates $s$~and~$\theta$.
Here [paper of~1825] he begins by proving the theorem, that the sum of the three
angles of a triangle, which is formed by shortest lines upon an arbitrary curved surface,
differs from~$180°$ by the area of the triangle, which corresponds to it in the representation
by means of parallel normals upon the auxiliary sphere of unit radius. From
this, by means of simple geometrical considerations, he derives the fundamental theorem,
that \Chg{``}{`}in the transformation of surfaces by bending, the measure of curvature at
every point remains unchanged.\Chg{''}{'} But there [paper of~1827] he first shows, in \Art[1827]{11},
that the measure of curvature can be expressed simply by means of the three
quantities $E$,~$F$,~$G$, and their derivatives with respect to $p$~and~$q$, from which follows
the theorem concerning the invariant property of the measure of curvature as a corollary;
and only much later, in \Art[1827]{20}, quite independently of this, does he prove the
theorem concerning the sum of the angles of a geodesic triangle.'' \\
\null\hfill Remark by Stäckel, Gauss's Works, vol.~\textsc{viii}, p.~443.

\LineRef[32]{3}{Art.~3, p.~84, l.~9}. $\cos^{2}\phi$, etc., is used here where the German text has~$\cos\phi^{2}$,~etc.

\LineRef[33]{3}{Art.~3, p.~84, l.~13}. $p^{2}$,~etc., is used here where the German text has~$pp$,~etc.

\LineRef[34]{7}{Art.~7, p.~89, ll.~13,~21}. Since $\lambda L$ is less than~$90°$, $\cos\lambda L$~is always positive
and, therefore, the algebraic sign of the expression for the volume of this pyramid
depends upon that of~$\sin L'L''$. Hence it is positive, zero, or negative according as
the arc~$L'L''$ is less than, equal to, or greater than~$180°$.

\LineRef[34]{7}{Art.~7, p.~89, ll.~14--21}. As is seen from the paper of~1827 (see \Pageref{6}), Gauss
\PageSep{112}
corrected this statement. To be correct it should read: for which we can write also,
according to well known principles of spherical trigonometry,
\[
\sin LL'·\sin L'·\sin L'L''
  = \sin L'L''·\sin L''·\sin L''L
  = \sin L''L·\sin L·\sin LL',
\]
if $L$,~$L'$,~$L''$ denote the three angles of the spherical triangle, where $L$~is the angle
measured from the arc~$LL''$ to~$LL'$, and so for the other angles. At the same time
we easily see that this value is one-sixth of the pyramid whose angular points are
the centre of the sphere and the three points $L$,~$L'$,~$L''$; and this pyramid is \emph{positive}
when the points $L$,~$L'$,~$L''$ are arranged in the same order about this triangle as the
points $(1)$,~$(2)$,~$(3)$ about the triangle $(1)\ (2)\ (3)$.

\LineRef{8}{Art.~8, p.~90, l.~7~fr.~bot}. In the German text $V$~stands for~$f$ in this equation
and in the next line but one.

\LineRef[35]{11}{Art.~11, p.~93, l.~8~fr.~bot}. In the German text, in the expression for~$B$, $(\alpha\beta' + \alpha\beta')$
stands for~$(\alpha'\beta + \alpha\beta')$.

\LineRef[36]{11}{Art.~11, p.~94, l.~17}. The vertices of the triangle are $M$,~$M'$,~$(3)$, whose coordinates
are $\alpha$,~$\beta$,~$\gamma$; $\alpha'$,~$\beta'$,~$\gamma'$; $0$,~$0$,~$1$, respectively. The pole of the arc~$MM'$ on
the same side as~$(3)$ is~$L$, whose coordinates are $X$,~$Y$,~$Z$. Now applying the formula
%[** TN: Omitted incorrect line number reference]
on \Pageref{89},\Chg{ line~10,}{}
\[
x'y'' - y'x'' = \sin L'L''\cos\lambda(3),
\]
to this triangle, we obtain
\[
\alpha\beta' - \beta\alpha' = \sin MM' \cos L(3)
\]
or, since
\[
MM' = 90°,\quad\text{and}\quad \cos L(3) = ±Z
\]
we have
\[
\alpha\beta' - \beta\alpha' = ±Z.
\]

\LineRef[37]{14}{Art.~14, p.~100, l.~19}. Here $X$,~$Y$,~$Z$; $\xi$,~$\eta$,~$\zeta$; $0$,~$0$,~$1$ take the place of $x$,~$y$,~$z$;
$x'$,~$y'$,~$z'$; $x''$,~$y''$,~$z''$ of the top of \Pageref{89}. Also $(3)$,~$\lambda$ take the place of $L'$,~$L''$, and
$\phi$~is the angle~$L$ in the note at the top of this page.

\LineRef[38]{14}{Art.~14, p.~101, l.~2~fr.~bot}. In the German text $\{\zeta X - \eta XYZ + \xi Y^{2}Z\}$ stands
for $\{\zeta X + \eta XYZ - \xi Y^{2}Z\}$.

\LineRef[39]{15}{Art.~15, p.~102, l.~13 and the following}. Transforming to polar coordinates,
$r$,~$\theta$,~$\psi$, by the substitutions (since on the auxiliary sphere $r = 1$)
\begin{gather*}
X = \sin\theta \sin\psi,\quad
Y = \sin\theta \cos\psi,\quad
Z = \cos\theta, \\
dX =  \sin\theta \cos\psi\, d\psi + \cos\theta \sin\psi\, d\theta,\qquad
dY = -\sin\theta \sin\psi\, d\psi + \cos\theta \cos\psi\, d\theta, \\
\Tag{(1)}
\Typo{=}{-}\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX)\quad\text{becomes}\quad
\cos\theta\, d\psi.
\end{gather*}
\PageSep{113}

In the figures on \Pgref{fig:102}, $PL$~and~$P'L'$ are arcs of great circles intersecting in
the point~$(3)$, and the element~$LL'$, which is not necessarily the arc of a great circle,
corresponds to the element of the geodesic line on the curved surface. $(2)PP'(1)$
also is the arc of a great circle. Here $P'P = d\psi$, $Z = \cos\theta ={}$Altitude of the zone
of which $LL'P'P$~is a part. The area of a zone varies as the altitude of the zone.
Therefore, in the case under consideration,
\[
\frac{\text{Area of zone}}{2\pi} = \frac{Z}{1}.
\]
Also
\[
\frac{\Area LL'P'P}{\text{Area of zone}} = \frac{d\psi}{2\pi}.
\]
From these two equations,
\[
\Tag{(2)}
\Area LL'P'P = Z\, d\psi,\quad\text{or}\quad \cos\theta\, d\psi.
\]
From (1)~and~(2)
\[
-\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX) = \Area LL'P'P.
\]

\LineRef[40]{15}{Art.~15, p.~102}. The point~$(3)$ in the figures on this page was added by the
translators.

\LineRef[41]{15}{Art.~15, p.~103, ll.~6--9}. It has been shown that $d\phi = \Area LL'P'P, = dA$, say.
Then
\[
\int_{\phi}^{\phi'} d\phi = \int_{0}^{A} dA,
\]
or
\[
\phi' - \phi = A,\quad\text{the finite area $LL'P'P$}.
\]

\LineRef[42]{15}{Art.~15, p.~103, l.~10 and the following}. Let $A$,~$B'$,~$B_{1}$ be the vertices of a
geodesic triangle on the curved surface, and let the corresponding triangle on the
auxiliary sphere be~$LL'L'_{1}L$, whose sides are not necessarily arcs of great circles. Let
$A$,~$B'$,~$B_{1}$ denote also the angles of the geodesic triangle. Here $B'$~is the supplement
of the angle denoted by~$B$ on \Pageref{103}. Let $\phi$~be the angle on the sphere
between the great circle arcs $L\lambda$,~$L(3)$, \ie, $\phi = (3)L\lambda$, $\lambda$~corresponding to the direction
of the element at~$A$ on the geodesic line~$AB'$, and let $\phi' = (3)L'\lambda_{1}$, $\lambda_{1}$~corresponding
to the direction of the element at~$B'$ on the line~$AB'$. Similarly, let $\psi = (3)L\mu$,
\PageSep{114}
$\psi' = (3)L'_{1}\mu_{1}$, $\mu$,~$\mu_{1}$ denoting the directions of the elements at
$A$,~$B_{1}$, respectively, on the line~$AB_{1}$. And let $\chi = (3)L'\nu$,
$\chi_{1} = (3)L'_{1}\nu_{1}$, $\nu$,~$\nu_{1}$ denoting the directions of the elements at
$B'$,~$B_{1}$, respectively, on the line~$B'B_{1}$.

Then from the first formula on \Pageref{103},
\begin{gather*}
\begin{aligned}[b]
\phi' - \phi &= \Area LL'P'P, \\
\psi' - \psi &= \Area LL'_{1}P'_{1}P, \\
\chi_{1} - \chi &= \Area L'L'_{1}P'_{1}P',
\end{aligned}
\qquad\qquad
%[Illustration]
\Graphic{1.5in}{114} \\
\psi' - \psi - (\phi' - \phi) - (\chi_{1} - \chi)
  = \Area L'L'_{1}P'_{1}P'
  - \Area LL'P'P
  - \Area L'L'_{1}P'_{1}P',
\end{gather*}
or
\[
\Tag{(1)}
(\phi - \psi) + (\chi - \phi') + (\psi' - \chi_{1})
  = \Area LL'_{1}L'L.
\]

Since $\lambda$,~$\mu$ represent the directions of the linear elements at~$A$ on the geodesic
lines $AB'$,~$AB_{1}$ respectively, the absolute value of the angle~$A$ on the surface is measured
by the arc~$\mu\lambda$, or by the spherical angle~$\mu L\lambda$. But $\phi - \psi = (3)L\lambda - (3)L\mu
= \mu L\lambda$. \\
Therefore
\[
A = \phi - \psi.
\]
Similarly
\begin{align*}
180° - B' &= -(\chi - \phi'), \\
B_{1} &= \psi' - \chi_{1}.
\end{align*}
Therefore, from~(1),
\[
A + B' + B_{1} - 180° = \Area LL'_{1}L'L.
\]

\LineRef[43]{15}{Art.~15, p.~103, l.~19}. In the German text $LL'P'P$ stands for~$LL'_{1}P'_{1}P$,
which represents the angle~$\psi' - \psi$.

\LineRef[44]{15}{Art.~15, p.~104, l.~12}. This general theorem may be stated as follows:

The sum of all the angles of a polygon of $n$~sides, which are shortest lines
upon the curved surface, is equal to the sum of $(n - 2)180°$ and the area of the
polygon upon the auxiliary sphere whose boundary is formed by the points~$L$ which
correspond to the points of the boundary of the given polygon, and in such a manner
that the area of this polygon may be regarded positive or negative according as it is
enclosed by its boundary in the same sense as the given figure or the contrary.

\LineRef[45]{16}{Art.~16, p.~104, l.~12~fr.~bot}. The \emph{zenith} of a point on the surface is the corresponding
point on the auxiliary sphere. It is the spherical representation of the
point.

\LineRef[46]{18}{Art.~18, p.~110, l.~10}. The normal to the surface is here taken in the direction
opposite to that given by~[9] \Pageref{107}.
\PageSep{115}
\BackMatter
%[** TN: Print "BIBLIOGRAPHY" title page]
\BibliographyPage
\PageSep{116}
%[Blank page]
\PageSep{117}


\begin{Bibliography}{%
This bibliography is limited to books, memoirs, etc., which use Gauss's method and which treat, more or less
generally, one or more of the following subjects: curvilinear coordinates, geodesic and isometric lines, curvature of
surfaces, deformation of surfaces, orthogonal systems, and the general theory of surfaces. Several papers which lie
beyond these limitations have been added because of their importance or historic interest. For want of space, generally,
papers on minimal surfaces, congruences, and other subjects not mentioned above have been excluded.

Generally, the numbers following the volume number give the pages on which the paper is found.

C.~R. will be used as an abbreviation for Comptes Rendus hebdomadaires des séances de l'Académie des
Sciences\Typo{.}{,} Paris.}

\Author{Adam, Paul} \Title{Sur les systèmes triples orthogonaux.} Thesis.
80~pp.\Add{,} Paris, 1887.

\Title{Sur les surfaces isothermiques à lignes de courbure
planes dans un système ou dans les deux systèmes.}
Ann.\ de l'École Normale, ser.~3, vol.~10, 319--358, 1893;
C.~R., vol.~116, 1036--1039, 1893.

\Title{Sur les surfaces admettant pour lignes de courbure
deux séries de cercles géodésiques orthogonaux.} Bull.\
de~la Soc.\ Math.\ de France, vol.~22, 110--115, 1894.

\Title{Mémoire sur la déformation des surfaces.} Bull.\ de~la
Soc.\ Math.\ de France, vol.~23, 219--240, 1895.

\Title{Sur la déformation des surfaces.} Bull.\ de~la Soc.\
Math.\ de France, vol.~23, 106--111, 1895; C.~R., vol.\
121, 551--553, 1895.

\Title{Sur la déformation des surfaces avec conservation des
lignes de courbure.} Bull.\ de~la Soc.\ Math.\ de France,
vol.~23, 195--196, 1895.

\Title{Théorème sur la déformation des surfaces de translation.}
Bull.\ de~la Soc.\ Math.\ de France, vol.~23, 204--209,
1895.

\Title{Sur un problème de déformation.} Bull.\ de~la Soc.\
Math.\ de France, vol.~24, 28--39, 1896.

\Author{Albeggiani, L.} \Title{Linee geodetiche tracciate sopra taluni superficie.}
Rend.\ del Circolo Mat.\ di Palermo, vol.~3, 80--119,
1889.

\Author{Allé, M.} \Title{Zur Theorie des Gauss'schen Krümmungsmaasses.}
Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften zu Wien,
vol.~74, 9--38, 1876.

\Author{Aoust, L. S. X. B.} \Title{Des coordonnées curvilignes se coupant
sous un angle quelconque.} Journ.\ für Math., vol.~58,
352--368, 1861.

\Title{Théorie géométrique des coordonnées curvilignes quelconques.}
C.~R., vol.~54, 461--463, 1862.

\Title{Sur la courbure des surfaces.} C.~R., vol.~57, 217--219,
1863.

%\Author{Aoust, L. S. X. B.}
\Title{Théorie des coordonnées curvilignes
quelconques.} Annali di Mat., vol.~6, 65--87, 1864; ser.~2,
vol.~2, 39--64, vol.~3, 55--69, 1868--69; ser.~2, vol.~5,
261--288, 1873.

\Author{August, T.} \Title{Ueber Flächen mit gegebener Mittelpunktsfläche
und über Krümmungsverwandschaft.} Archiv
der Math.\ und Phys., vol.~68, 315--352, 1882.

\Author{Babinet.} \Title{Sur la courbure des surfaces.} C.~R., vol.~49, 418--424,
1859.

\Author{Bäcklund, A. V.} \Title{Om ytar med konstant negativ kröking.}
Lunds Univ.\ Årsskrift, vol.~19, 1884.

\Author{Banal, R.} \Title{Di una classe di superficie a tre dimensioni a
curvatura totale nulla.} Atti del Reale Instituto Veneto,
ser.~7, vol.~6, 998--1004, 1895.

\Author{Beliankén, J.} \Title{Principles of the theory of the development
of surfaces. Surfaces of constant curvature.} \Chg{(Russian).}{(Russian.)}
Kief Univ.\ Reports, Nos.\ 1~and~3; and Kief, \Chg{pp.\ \textsc{ii}~+~129}{\textsc{ii}~+~129~pp.},
1898.

\Author{Beltrami, Eugenio.} \Title{Di alcune formole relative alla curvatura
delle superficie.} Annali di Mat., vol\Add{.}~4, 283--284,
1861.

\Title{Richerche di analisi applicata alla geometria.} Giornale
di Mat., vol.~2, 267--282, 297--306, 331--339, 355--375,
1864; vol.~3, 15--22, 33--41, 82--91, 228--240, 311--314, 1865.

\Title{Delle variabili complesse sopra una superficie qualunque.}
Annali di Mat., ser.~2, vol.~1, 329--366, 1867.

\Title{Sulla teorica generale dei parametri differenziali.}
Mem.\ dell'Accad.\ di Bologna, ser.~2, vol.~8, 549--590,
1868.

\Title{Sulla teoria generale delle superficie.} Atti dell'Ateneo
Veneto, vol.~5, 1869.

\Title{Zur Theorie des Krümmungsmaasses.} Math.\ Annalen,
vol.~1, 575--582, 1869.

\Author{Bertrand, J.} \Title{Mémoire sur la théorie des surfaces.} Journ.\
de Math., vol.~9, 133--154, 1844.
\PageSep{118}

\Author{Betti, E.} \Title{Sopra i sistemi di superficie isoterme e orthogonali.}
Annali di Mat., ser.~2, vol.~8, 138--145, 1877.

\Author{Bianchi, Luigi.} \Title{Sopra la deformazione di una classe di
superficie.} Giornale di Mat., vol.~16, 267--269, 1878.

\Title{Ueber die Flächen mit constanter negativer Krümmung.}
Math.\ Annalen, vol.~16, 577--582, 1880.

\Title{Sulle superficie a curvatura costante positiva.} Giornale
di Mat., vol.~20, 287--292, 1882.

\Title{Sui sistemi tripli cicilici di superficie orthogonali.}
Giornale di Mat., vol.~21, 275--292, 1883; vol.~22, 333--373,
1884.

\Title{Sopra i sistemi orthogonali di Weingarten.} Atti della
Reale Accad.\ dei Lincei, ser.~4, vol.~1, 163--166, 243--246,
1885; Annali di Mat., ser.~2, vol.~13, 177--234,
1885, and ser.~2, vol.~14, 115--130, 1886.

\Title{Sopra una classe di sistemi tripli di superficie orthogonali,
che contengono un sistema di elicoidi aventi a
comune l'asse ed il passo.} Annali di Mat., ser.~2, vol.~13,
39--52, 1885.

\Title{Sopra i sistemi tripli di superficie orthogonali che contengono
un sistema di superficie pseudosferiche.} Atti
della Reale Accad.\ dei Lincei, ser.~4, vol.~2, 19--22,
1886.

\Title{Sulle forme differenziali quadratiche indefinite.} Atti
della Reale Accad.\ dei Lincei, vol.~$4_{2}$, 278, 1888; Mem.\
della Reale Accad.\ dei Lincei, ser.~4, vol.~5, 539--603,
1888.

\Title{Sopra alcune nuove classi di superficie e di sistemi
tripli orthogonali.} Annali di Mat., ser.~2, vol.~18, 301--358,
1890.

\Title{Sopra una nuova classe di superficie appartenenti a
sistemi tripli orthogonali.} Atti della Reale Accad.\ dei
Lincei, ser.~4, vol.~$6_{1}$, 435--438, 1890.

\Title{Sulle superficie i cui piani principali hanno costante
il rapporto delle distanze da un punto fisso.} Atti
della Reale Accad.\ dei Lincei, ser.~5, vol.~$3_{2}$, 77--84,
1894.

\Title{Sulla superficie a curvatura nulla negli spazi curvatura
costante.} Atti della Reale Accad.\ di Torino, vol.~30,
743--755, 1895.

\Title{Lezioni di geometria differenziale.} \textsc{viii}~+~541~pp.\Add{,}
Pisa, 1894. Translation into German by Max Lukat,
\Title{Vorlesungen über Differentialgeometrie.} \textsc{xvi}~+~659~pp.\Add{,}
Leipzig, 1896--99.

\Title{Sopra una classe di superficie collegate alle superficie
pseudosferiche.} Atti della Reale Accad.\ dei Lincei, ser.~5,
vol.~$5_{1}$, 133--137, 1896.

\Title{Nuove richerche sulle superficie pseudosferiche.} Annali
di Mat., ser.~2, vol.~24, 347--386, 1896.

\Title{Sur deux classes de surfaces qui engendrent par un
mouvement hélicoidal une famille de~Lamé.} Ann.\
Faculté des sci.\ de Toulouse, vol.~11~H, 1--8, 1897.

\Author{Bianchi, Luigi.} \Title{Sopra le superficie a curvatura costante
positiva.} Atti della Reale Accad.\ dei Lincei, ser.~5,
vol.~$8_{1}$, 223--228, 371--377, 484--489, 1899.

\Title{Sulla teoria delle transformazioni delle superficie a
curvatura costante.} Annali di Mat., ser.~3, vol.~3, 185--298,
1899.

\Author{Blutel, E.} \Title{Sur les surfaces à lignes de courbure sphérique.}
C.~R., vol.~122, 301--303, 1896.

\Author{Bonnet, Ossian.} \Title{Mémoire sur la théorie des surfaces isothermes
orthogonales.} \Chg{Jour.}{Journ.}\ de l'École Polyt., cahier~30,
vol.~18, 141--164, 1845.

\Title{Sur la théorie générale des surfaces.} Journ.\ de l'École
Polyt., cahier~32, vol.~19, 1--146, 1848; C.~R., vol.~33,
89--92, 1851; vol.~37, 529--532, 1853.

\Title{Sur les lignes géodésiques.} C.~R., vol.~41, 32--35,
1855.

\Title{Sur quelques propriétés des lignes géodésiques.} C.~R.,
vol.~40, 1311--1313, 1855.

\Title{Mémoire sur les surfaces orthogonales.} C.~R., vol.~54,
554--559, 655--659, 1862.

\Title{Démonstration du théorème de Gauss relatif aux petits
triangles géodésiques situés sur une surface courbe quelconque.}
C.~R., vol.~58, 183--188, 1864.

\Title{Mémoire sur la théorie des surfaces applicables sur
une surface donnée.} Journ.\ de l'École Polyt., cahier~41,
vol.~24, 209--230, 1865; cahier~42, vol.~25, 1--151,
1867.

\Title{Démonstration des propriétés fondamentales du système
de coordonnées polaires géodésiques.} C.~R., vol.~97,
1422--1424, 1883.

\Author{Bour, Edmond.} \Title{Théorie de~la déformation des surfaces.}
Journ.\ de l'École Polyt., cahier~39, vol.~22, 1--148,
1862.

\Author{Brill, A.} \Title{Zur Theorie der geodätischen Linie und des
geodätischen Dreiecks.} Abhandl.\ der Kgl.\ Gesell.\ der
Wissenschaften zu München, vol.~14, 111--140, 1883.

\Author{Briochi, Francesco.} \Title{Sulla integrazione della equazione della
geodetica.} Annali di sci.\ Mat.\ e Fis., vol.~4, 133--135,
1853.

\Title{Sulla teoria delle coordinate curvilinee.} Annali di
Mat., ser.~2, vol.~1, 1--22, 1867.

\Author{Brisse, C.} \Title{Exposition analytique de~la théorie des surfaces.}
Ann.\ de l'École Normale, ser.~2, vol.~3, 87--146, 1874;
Journ.\ de l'École Polyt., cahier~53, 213--233, 1883.

\Author{Bukrejew, B.} \Title{Surface elements of the surface of constant
curvature.} \Chg{(Russian).}{(Russian.)} Kief Univ.\ Reports, No.~7,
4~pp., 1897.

\Title{Elements of the theory of surfaces.} \Chg{(Russian).}{(Russian.)} Kief
Univ.\ Reports, Nos.~1,~9, and~12, 1897--99.

\Author{Burali-Forti, C.} \Title{Sopra alcune questioni di geometria differenziale.}
Rend.\ del Circolo Mat.\ di Palermo, vol.~12,
111--132, 1898.
\PageSep{119}

\Author{Burgatti, P.} \Title{Sulla torsione geodetica delle linee tracciate
sopra una superficie.} Rend.\ del Circolo Mat.\ di Palermo,
vol.~10, 229--240, 1896.

\Author{Burnside, W.} \Title{The lines of zero length on a surface as
curvilinear coordinates.} Mess.\ of Math., ser.~2, vol.~19,
99--104, 1889.

\Author{Campbell, J.} \Title{Transformations which leave the lengths of
arcs on surfaces unaltered.} Proceed.\ London Math.\
Soc., vol.~29, 249--264, 1898.

\Author{Carda, K.} \Title{Zur Geometrie auf Flächen constanter Krümmung.}
Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften
zu Wien, vol.~107, 44--61, 1898.

\Author{Caronnet, Th.} \Title{Sur les centres de courbure géodésiques.}
C.~R., vol.~115, 589--592, 1892.

\Title{Sur des couples de surfaces applicables.} Bull.\ de~la
Soc.\ Math.\ de France, vol.~21, 134--140, 1893.

\Title{Sur les surfaces à lignes de courbure planes dans les deux
systèmes et isothermes.} C.~R., vol.~116, 1240--1242, 1893.

\Title{Recherches sur les surfaces isothermiques et les surfaces
dont rayons de courbure sont fonctions l'un de
l'autre.} Thesis, 66~pp.\Add{,} Paris, 1894.

\Author{Casorati, Felice.} \Title{Nuova definizione della curvatura delle
superficie e suo confronto con quella di Gauss.} Reale
Istituto Lombardo di sci.\ e let., ser.~2, vol.~22, 335--346,
1889.

\Title{Mesure de~la courbure des surfaces suivant l'idee commune.
Ses rapports avec les mesures de courbure Gaussienne
et moyenne.} Acta Matematica, vol.~14, 95--110, 1890.

\Author{Catalan, E.} \Title{Mémoire sur les surfaces dont les rayons de
courbure en chaque point sont égaux et de signes contraires.}
Journ.\ de l'École Polyt., cahier~37, vol.~21, 130--168,
1858; C.~R., vol.~41, 35--38, 274--276, 1019--1023, 1855.

\Author{Cayley, Arthur.} \Title{On the Gaussian theory of surfaces.} Proceed.\
London Math.\ Soc., vol.~12, 187--192, 1881.

\Title{On the geodesic curvature of a curve on a surface.}
Proceed.\ London Math.\ Soc., vol.~12, 110--117, 1881.

\Title{On some formulas of Codazzi and Weingarten in relation
to the application of surfaces to each other.} Proceed.\
London Math.\ Soc., vol.~24, 210--223, 1893.

\Author{Cesàro, E.} \Title{Theoria intrinseca delle deformazioni infinitesime.}
Rend.\ dell'Accad.\ di Napoli, ser.~2, vol.~8, 149--154,
1894.

\Author{Chelini, D.} \Title{Sulle formole fondamentali risguardanti la curvatura
delle superficie e delle linee.} Annali di Sci.\
Mat.\ e Fis., vol.~4, 337--396, 1853.

\Title{Della curvatura delle superficie, con metodo diretto ed
intuitivo.} Rend.\ dell'Accad.\ di Bologna, 1868, 119;
Mem.\ dell'Accad.\ di Bologna, ser.~2, vol.~8, 27, 1868.

\Title{Teoria delle coordinate curvilinee nello spazio e nelle
superficie.} Mem.\ dell'Accad.\ di Bologna, ser.~2, vol.~8,
483--533, 1868.

\Author{Christoffel, Elwin.} \Title{Allgemeine Theorie der geodätische
Dreiecke.} Abhandl.\ der Kgl.\ Akad.\ der Wissenschaften
zu Berlin, 1868, 119--176.

\Author{Codazzi, Delfino.} \Title{Sulla teorica delle coordinate curvilinee e
sull uogo de'centri di curvatura d'una superficie qualunque.}
Annali di sci.\ Mat.\ e Fis., vol.~8, 129--165,
1857.

\Title{Sulle coordinate curvilinee d'una superficie e dello
spazio.} Annali di Mat., ser.~2, vol.~1, 293--316; vol.~2,
101--119, 269--287; vol.~4, 10--24; vol.~5, 206--222; 1867--1871.

\Author{Combescure, E.} \Title{Sur les déterminants fonctionnels et les
coordonnèes curvilignes.} Ann.\ de l'École Normale, ser\Add{.}~1,
vol.~4, 93--131, 1867.

\Title{Sur un point de~la théorie des surfaces.} C.~R., vol.~74,
1517--1520, 1872.

\Author{Cosserat, E.} \Title{Sur les congruences des droites et sur la théorie
des surfaces.} Ann.\ Faculté des sci.\ de Toulouse, vol.~7~N,
1--62, 1893.

\Title{Sur la déformation infinitésimale d'une surface flexible
et inextensible et sur les congruences de droites.} Ann.\
Faculté des sci.\ de Toulouse, vol.~8~E, 1--46, 1894.

\Title{Sur les surfaces rapportées à leurs lignes de longeur
nulle.} C.~R., vol.~125, 159--162, 1897.

\Author{Craig, T.} \Title{Sur les surfaces à lignes de courbure isométriques.}
C.~R., vol.~123, 794--795, 1896.

\Author{Darboux, Gaston.} \Title{Sur les surfaces orthogonales.} Thesis,
45~pp.\Add{,} Paris, 1866.

\Title{Sur une série de lignes analogues aux lignes géodésiques.}
Ann.\ de l'École Normale, vol.~7, 175--180, 1870.

\Title{Mémoire sur la théorie des coordonnées curvilignes et
des systèmes orthogonaux.} Ann.\ de l'École Normale,
ser.~2, vol.~7, 101--150, 227--260, 275--348, 1878.

\Title{Sur les cercles géodésiques.} C.~R., vol.~96, 54--56,
1883.

\Title{Sur les surfaces dont la courbure totale est constante.
Sur les surfaces à courbure constante. Sur l'équation
aux dérivées partielles des surfaces à courbure constante.}
C.~R., vol.~97, 848--850, 892--894, 946--949, 1883.

\Title{Sur la représentation sphérique des surfaces.} C.~R.,
vol.~68, 253--256, 1869; vol.~94, 120--122, 158--160, 1290--1293,
1343--1345, 1882; vol.~96, 366--368, 1883; Ann.\
de l'École Normale, ser.~3, vol.~5, 79--96, 1888.

\Title{Leçons sur la théorie générale des surfaces et les applications
géométriques du calcul infinitésimale.} 4~vols.
Paris, 1887--1896.

\Title{Sur les surfaces dont la courbure totale est constante.}
Ann.\ de l'École Normale, ser.~3, vol.~7, 9--18, 1890.

\Title{Sur une classe remarkable de courbes et de surfaces
algebriques.} Second edition. Paris, 1896.

\Title{Leçons sur les systèmes orthogonaux et les coordonnées
curvilignes.} Vol.~1. Paris, 1898.
\PageSep{120}

\Author{Darboux, Gaston.} \Title{Sur les transformations des surfaces à courbure
totale constante.} C.~R., vol.~128, 953--958, 1899.

\Title{Sur les surfaces à courbure constante positive.} C.~R.,
vol.~128, 1018--1024, 1899.

\Author{Demartres, G.} \Title{Sur les surfaces réglées dont l'\Typo{element}{élément} linéaire
est réductible à la forme de Liouville.} C.~R., vol.~110,
329--330, 1890.

\Author{Demoulin, A.} \Title{Sur la correspondence par orthogonalité des
éléments.} C.~R., vol.~116, 682--685, 1893.

\Title{Sur une propriété caractéristique de l'\Typo{element}{élément} linéaire
des surfaces de révolution.} Bull.\ de~la Soc.\ Math.\ de
France, vol.~22, 47--49, 1894.

\Title{Note sur la détermination des couples de surfaces
applicables telles que la distance de deux points correspondants
soit constante.} Bull.\ de~la Soc.\ Math.\ de
France, vol.~23, 71--75, 1895.

\Author{de Salvert}, see (de) Salvert.

\Author{de Tannenberg}, see (de) Tannenberg.

\Author{Dickson, Benjamin.} \Title{On the general equations of geodesic
lines and lines of curvature on surfaces.} Camb.\ and
Dub.\ Math.\ Journal, vol.~5, 166--171, 1850.

\Author{Dini, Ulisse.} \Title{Sull'equazione differenzialle delle superficie
applicabili su di una superficie data.} Giornale di Mat.,
vol.~2, 282--288, 1864.

\Title{Sulla teoria delle superficie.} Giornale di Mat., vol.~3,
65--81, 1865.

\Title{Ricerche sopra la teorica delle superficie.} Atti della
Soc.\ Italiana dei~XL\@. Firenze, 1869.

\Title{Sopra alcune formole generali della teoria delle superficie
e loro applicazioni.} Annali di Mat., ser.~2, vol.~4,
175--206, 1870.

\Author{van Dorsten, R.} \Title{Theorie der Kromming von lijnen op
gebogen oppervlakken.} Diss.\ Leiden.\ Brill. 66~pp.\Add{,} 1885.

\Author{Egorow, D.} \Title{On the general theory of the correspondence of
surfaces.} (Russian.) Math.\ Collections, pub.\ by Math.\
Soc.\ of Moscow, vol.~19, 86--107, 1896.

\Author{Enneper, A.} \Title{Bemerkungen zur allgemeinen Theorie der
Flächen.} Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften
zu Göttingen, 1873, 785--804.

\Title{Ueber ein geometrisches Problem.} Nachr.\ der Kgl.\
Gesell.\ der Wissenschaften zu Göttingen, 1874, 474--485.

\Title{Untersuchungen über orthogonale Flächensysteme.}
Math.\ Annalen, vol.~7, 456--480, 1874.

\Title{Bemerkungen über die Biegung einiger Flächen.}
Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
1875, 129--162.

\Title{Bemerkungen über einige Flächen mit constantem
Krümmungsmaass.} Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften
zu Göttingen, 1876, 597--619.

\Title{Ueber die Flächen mit einem system sphärischer
Krümmungslinien.} Journ.\ für Math., vol.~94, 829--341,
1883.

%\Author{Enneper, A.}
\Title{Bemerkungen über einige Transformationen
von Flächen.} Math.\ Annalen, vol.~21, 267--298, 1883.

\Author{Ermakoff, W.} \Title{On geodesic lines.} (Russian.) Math.\ Collections,
pub.\ by Math.\ Soc.\ of Moscow, vol.~15, 516--580,
1890.

\Author{von Escherich, G.} \Title{Die Geometrie auf den Flächen constanter
negativer Krümmung.} Sitzungsb.\ der Ksl.\
Akad.\ der Wissenschaften zu Wien, vol.~69, part~II,
497--526, 1874.

\Title{Ableitung des allgemeinen Ausdruckes für das Krümmungsmaass
der Flächen.} Archiv für Math.\ und
Phys., vol.~57 385--392, 1875.

\Author{Fibbi, C.} \Title{Sulle superficie che contengono un sistema di
geodetiche a torsione costante.} Annali della Reale
Scuola Norm.\ di Pisa, vol.~5, 79--164, 1888.

\Author{Firth, W.} \Title{On the measure of curvature of a surface referred
to polar coordinates.} Oxford, Camb., and Dub.\ Mess.,
vol.~5, 66--76, 1869.

\Author{Fouché, M.} \Title{Sur les systèmes des surfaces triplement orthogonales
où les surfaces d'une même famille admettent la
même représentation sphérique de leurs lignes de courbure.}
C.~R., vol.~126, 210--213, 1898.

\Author{Frattini, G.} \Title{Alcune formole spettanti alla teoria infinitesimale
delle superficie.} Giornale di Mat., vol.~13, 161--167,
1875.

\Title{Un esempio sulla teoria delle coordinate curvilinee
applicata al calcolo integrale.} Giornale di Mat., vol.~15,
1--27, 1877.

\Author{Frobenius, G.} \Title{Ueber die in der Theorie der Flächen auftretenden
Differentialparameter.} Journ.\ für Math., vol.~110,
1--36, 1892.

\Author{Gauss, K. F.} \Title{Allgemeine Auflösung der Aufgabe: Die
Theile einer gegebenen Fläche auf einer anderen gegebenen
Fläche so abzubilden, dass die Abbildung dem
Abgebildeten in den kleinsten Theilen ähnlich wird.}
Astronomische Abhandlungen, vol.~3, edited hy H.~C.
Schumacher, Altona, 1825. The same, Gauss's Works,
vol.~4, 189--216, 1880; Ostwald's Klassiker, No.~55,
edited by A.~Wangerin, 57--81, 1894.

\Author{Geiser, C. F.} \Title{Sur la théorie des systèmes triples orthogonaux.}
Bibliothèque universelle, Archives des sciences, ser.~4,
vol.~6, 363--364, 1898.

\Title{Zur Theorie der tripelorthogonalen Flächensysteme.}
Vierteljahrschrift der Naturf.\ Gesell.\ in Zurich, vol.~43,
317--326, 1898.

\Author{Germain, Sophie.} \Title{Mémoire sur la courbure des surfaces.}
Journ.\ für Math., vol.~7, 1--29, 1831.

\Author{Gilbert, P.} \Title{Sur l'emploi des cosinus directeurs de~la normale
dans la théorie de~la courbure des surfaces.} Ann.\
de~la Soc.\ sci.\ de Bruxelles, vol.~18~B, 1--24, 1894.

\Author{Genty, E.} \Title{Sur les surfaces à courbure totale constante.}
Bull.\ de~la Soc.\ Math.\ de France, vol.~22, 106--109, 1894.
\PageSep{121}

\Author{Genty, E.} \Title{Sur la déformation infinitésimale de surfaces.}
Ann.\ de~la Faculté des sci.\ de Toulouse, vol.~9~E, 1--11,
1895.

\Author{Goursat, E.} \Title{Sur les systèmes orthogonaux.} C.~R., vol.~121,
883--884, 1895.

\Title{Sur les équations d'une surface rapportée à ses lignes
de longueur nulle.} Bull.\ de~la Soc.\ Math.\ de France,
vol.~26, 83--84, 1898.

\Author{Grassmann, H.} \Title{Anwendung der Ausdehnungslehre auf die
allgemeine Theorie der Raumcurven und krummen
Flächen.} Diss.\ Halle, 1893.

\Author{Guichard, C.} \Title{Surfaces rapportées à leur lignes asymptotiques
et congruences rapportées à leurs dévéloppables.}
Ann.\ de l'École Normale, ser.~3, vol.~6, 333--348, 1889.

\Title{Recherches sur les surfaces à courbure totale constante
et certaines surfaces qui s'y rattachent.} Ann.\ de l'École
Normale, ser.~3, vol.~7, 233--264, 1890.

\Title{Sur les surfaces qui possèdent un réseau de géodésiques
conjuguées.} C.~R., vol.~110, 995--997, 1890.

\Title{Sur la déformation des surfaces.} Journ.\ de Math.,
ser.~5, vol.~2, 123--215, 1896.

\Title{Sur les surfaces à courbure totale constante.} C.~R.,
vol.~126, 1556--1558, 1616--1618, 1898.

\Title{Sur les systémes orthogonaux et les systémes cycliques.}
Ann.\ de l'École Normale, ser.~3, vol.~14, 467--516, 1897;
vol.~15, 179--227, 1898.

\Author{Guldberg, Alf.} \Title{Om Bestemmelsen af de geodaetiske Linier
paa visse specielle Flader.} Nyt Tidsskrift for Math.\
Kjöbenhavn, vol.~6~B, 1--6, 1895.

\Author{Hadamard, J.} \Title{Sur les lignes géodésiques des surfaces spirales
et les équations différentielles qui s'y rapportent.} Procès
verbeaux de~la Soc.\ des sci.\ de Bordeaux, 1895--96, 55--58.

\Title{Sur les lignes géodésiques des surfaces à courbures
opposées.} C.~R., vol.~124, 1503--1505, 1897.

\Title{Les surfaces à courbures opposées et leurs lignes
géodésiques.} Journ.\ de Math., ser.~5, vol.~4, 27--73, 1898.

\Author{Haenig, Conrad.} \Title{Ueber Hansen's Methode, ein geodätisches
Dreieck auf die Kugel oder in die Ebene zu übertragen.}
Diss., 36~pp., Leipzig, 1888.

\Author{Hansen, P. A.} \Title{Geodätische Untersuchungen\Add{.}} Abhandl.\ der
Kgl.\ Gesell.\ der Wissenschaften zu \Typo{Leipsig}{Leipzig}, vol.~18,
1865; vol.~9, 1--184, 1868.

\Author{Hathaway, A.} \Title{Orthogonal surfaces.} Proc.\ Indiana Acad.,
1896, 85--86.

\Author{Hatzidakis, J. N.} \Title{Ueber einige Eigenschaften der Flächen
mit constantem Krümmungsmaass.} Journ.\ für Math.,
vol.~88, 68--73, 1880.

\Title{Ueber die Curven, welche sich so bewegen können,
dass sie stets geodätische Linien der von ihnen erzeugten
Flächen bleiben.} Journ.\ für Math., vol.~95, 120--139,
1883.

\Author{Hatzidakis, J. N.} \Title{Biegung mit Erhaltung der Hauptkrümmungsradien.}
Journ.\ für Math., vol.~117, 42--56,
1897.

\Author{Hilbert, D.} \Title{Ueber Flächen von constanter Gaussscher Krümmung.}
Trans.\ Amer. Math.\ Society, vol.~2, 87--99,
1901.

\Author{Hirst, T.} \Title{Sur la courbure d'une série de surfaces et de
lignes.} Annali di Mat., vol.~2, 95--112, 148--167, 1859.

\Author{Hoppe, R.} \Title{Zum Problem des dreifach orthogonalen Flächensystems.}
Archiv für Math.\ und Phys., vol.~55, 362--391,
1873; vol.~56, 153--163, 1874; vol.~57, 89--107, 255--277,
366--385, 1875; vol.~58, 37--48, 1875.

\Title{Principien der Flächentheorie.} Archiv für Math.\ und
Phys., vol.~59, 225--323, 1876; Leipzig, Koch, 179~pp.\Add{,}
1876.

\Title{Geometrische Deutung der Fundamentalgrössen zweiter
Ordnung der Flächentheorie.} Archiv für Math.\ und
Phys., vol.~60, 65--71, 1876.

\Title{Nachträge zur Curven- und Flächentheorie.} Archiv
für Math.\ und Phys., vol.~60, 376--404, 1877.

\Title{Ueber die kürzesten Linien auf den Mittelpunktsflächen.}
Archiv für Math.\ und Phys., vol.~63, 81--93,
1879.

\Title{Untersuchungen über kürzeste Linien.} Archiv für
Math.\ und Phys., vol.~64, 60--74, 1879.

\Title{Ueber die Bedingung, welcher eine Flächenschaar
genügen muss, um einen dreifach orthogonalen system
anzugehören.} Archiv für Math.\ und Phys., vol.~63,
285--294, 1879.

\Title{Nachtrag zur Flächentheorie.} Archiv für Math.\ und
Phys., vol.~68, 439--440, 1882.

\Title{Ueber die sphärische Darstellung der asymptotischen
Linien einer Fläche.} Archiv für Math.\ und Phys., ser.~2,
vol.~10, 443--446, 1891.

\Title{Eine neue Beziehung zwischen den Krümmungen von
Curven und Flächen.} Archiv für Math.\ und Phys.,
ser.~2, vol.~16, 112, 1898.

\Author{Jacobi, C. G. J.} \Title{Demonstratio et amplificatio nova theorematis
Gaussiani de quadratura integra trianguli in
data superficie e lineis brevissimis formati.} Journ.\ für
Math., vol.~16, 344--350, 1837.

\Author{Jamet, V.} \Title{Sur la théorie des lignes géodésiques.} Marseille
Annales, vol.~8, 117--128, 1897.

\Author{Joachimsthal, F.} \Title{Demonstrationes theorematum ad superficies
curvas spectantium.} Journ.\ für Math., vol.~30,
347--350, 1846.

\Title{Anwendung der Differential- und Integralrechnung
auf die allgemeine Theorie der Flächen und Linien
doppelter Krümmung.} Leipzig, Teubner, first~ed.,
1872; second~ed., 1881; third~ed., \textsc{x}~+~308~pp., revised
by L.~Natani, 1890.
\PageSep{122}

\Author{Knoblauch, Johannes.} \Title{Einleitung in die allegemeine Theorie
der krummen Flächen.} Leipzig, Teubner, \textsc{viii}~+~267~pp.,
1888.

\Title{Ueber Fundamentalgrössen in der Flächentheorie.}
Journ.\ für Math., vol.~103, 25--39, 1888.

\Title{Ueber die geometrische Bedeutung der flächentheoretischen
Fundamentalgleichungen.} Acta Mathematica,
vol.~15, 249--257, 1891.

\Author{Königs, G.} \Title{Résumé d'un mémoire sur les lignes géodésiques.}
Ann.\ Faculté des sci.\ de Toulouse, vol.~6~P, 1--34, 1892.

\Title{Une théorème de géométrie \Typo{infinitesimale}{infinitésimale}.} C.~R., vol.~116,
569, 1893.

\Title{Mémoire sur les lignes géodésiques.} Mém.\ présentés
par savants à l'Acad.\ des sci.\ de l'Inst.\ de France, vol.~31,
No.~6, 318~pp., 1894.

\Author{Kommerell, V.} \Title{Beiträge zur Gauss'schen Flächentheorie.}
Diss., \textsc{iii}~+~46~pp., Tübingen, 1890.

\Title{Eine neue Formel für die mittlere Krümmung und
das Krümmungsmaass einer Fläche.} Zeitschrift für
Math.\ und Phys., vol.~41, 123--126, 1896.

\Author{Köttfritzsch, Th.} \Title{Zur Frage über isotherme Coordinatensysteme.}
Zeitschrift für Math.\ und Phys., vol.~19, 265--270,
1874.

\Author{Kummer, E. E.} \Title{Allgemeine Theorie der geradlinigen
Strahlensysteme.} Journ.\ für Math., vol.~57, 189--230,
1860.

\Author{Laguerre.} \Title{Sur les formules fondamentales de~la théorie des
surfaces.} Nouv.\ Ann.\ de Math., ser.~2, vol.~11, 60--66,
1872.

\Author{Lamarle, E.} \Title{Exposé géométrique du calcul differential et
integral.} Chaps. \textsc{x}--\textsc{xiii}. Mém.\ couronnés et autr.\
mém.\ publ.\ par l'Acad.\ Royale de Belgique, vol.~15, 418--605,
1863.

\Author{Lamé, Gabriel.} \Title{Mémoire sur les coordonnées curvilignes.}
Journ.\ de Math., vol.~5, 313--347, 1840.

\Title{Leçons sur les coordonnées curvilignes.} Paris, 1859.

\Author{Lecornu, L.} \Title{Sur l'équilibre des surfaces flexibles et inextensibles.}
Journ.\ de l'École Polyt., cahier~48, vol.~29,
1--109, 1880.

\Author{Legoux, A.} \Title{Sur l'integration de l'équation $ds^{2} = E\, du^{2} +
2F\, du\, dv + G\, dv^{2}$.} Ann.\ de~la Faculté des sci.\ de
Toulouse, vol.~3~F, 1--2, 1889.

\Author{Lévy, L.} \Title{Sur les systèmes de surfaces triplement orthogonaux.}
Mém.\ couronnés et mém.\ des sav.\ publiés par
l'Acad.\ Royale de Belgique, vol.~54, 92~pp., 1896.

\Author{Lévy, Maurice.} \Title{Sur une transformation des coordonnées
curvilignes orthogonales et sur les coordonnées curvilignes
comprenant une famille quelconque de surfaces du
second ordre.} Thesis, 33~pp., Paris, 1867.

\Title{Mémoire sur les coordonnées curvilignes orthogonales.}
Journ.\ de l'École Polyt., cahier~43, vol.~26, 157--200,
1870.

%\Author{Lévy, Maurice.}
\Title{Sur une application industrielle du théorème
de Gauss relatif à la courbure des surfaces.} C.~R., vol.~86,
111--113, 1878.

\Author{Lie, Sophus.} \Title{Ueber Flächen, deren Krümmungsradien durch
eine Relation verknüpft sind.} Archiv for Math.\ og
Nat., Christiania, vol.~4, 507--512, 1879.

\Title{Zur Theorie der Flächen constanter Krümmung.}
Archiv for Math.\ og Nat., Christiania, vol.~4, 345--354,
355--366, 1879; vol.~5 282--306, 328--358, 518--541, 1881.

\Title{Untersuchungen über geodätische Curven.} Math.\
Annalen, vol.~20 357--454, 1882.

\Title{Zur Geometrie einer Monge'schen Gleichung.} Berichte
der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
vol.~50 1--2, 1898.

\Author{von Lilienthal, Reinhold.} \Title{Allgemeine Eigenschaften von
Flächen, deren Coordinaten sich durch reellen Teile
dreier analytischer Functionen einer complexen Veränderlichen
darstellen lassen.} Journ.\ für Math., vol.~98,
131--147, 1885.

\Title{Untersuchungen zur allgemeinen Theorie der krummen
Oberflächen und geradlinigen Strahlensysteme.}
Bonn, E.~Weber, 112~pp., 1886.

\Title{Zur Theorie der Krümmungsmittelpunktsflächen.}
Math.\ Annalen, vol.~30, 1--14, 1887.

\Title{Ueber die Krümmung der Curvenschaaren.} Math.\
Annalen, vol.~32, 545--565, 1888.

\Title{Zur Krümmungstheorie der Flächen.} Journ.\ für
Math., vol.~104, 341--347, 1889.

\Title{Zur Theorie des Krümmungsmaasses der Flächen.}
Acta Mathematica, vol.~16, 143--152, 1892.

\Title{Ueber geodätische Krümmung.} Math.\ Annalen,
vol.~42, 505--525, 1893.

\Title{Ueber die Bedingung, unter der eine Flächenschaar
einem dreifach orthogonalen Flächensystem angehört.}
Math.\ Annalen, vol.~44, 449--457, 1894.

\Author{Lipschitz, Rudolf.} \Title{Beitrag zur Theorie der Krümmung.}
Journ.\ für Math., vol.~81, 230--242, 1876.

\Title{Untersuchungen über die Bestimmung von Oberflächen
mit vorgeschriebenen, die Krümmungsverhältnisse
betreffenden Eigenschaften.} Sitzungsb.\ der Kgl.\ Akad.\
der Wissenschaften zu Berlin, 1882, 1077--1087; 1883,
169--188.

\Title{Untersuchungen über die Bestimmung von Oberflächen
mit vorgeschriebenem Ausdruck des Linearelements.}
Sitzungsb.\ der Kgl.\ Akad.\ der Wissenschaften
zu Berlin, 1883, 541--560.

\Title{Zur Theorie der krummen Oberflächen.} Acta Mathematica,
vol.~10, 131--136, 1887.

\Author{Liouville, Joseph.} \Title{Sur un théorème de M.~Gauss concernant
le produit des deux rayons de courbure principaux
en chaque point d'une surface.} Journ.\ de Math.,
vol.~12, 291--304, 1847.
\PageSep{123}

\Author{Liouville, Joseph.} \Title{Sur la théorie générale des surfaces.}
Journ.\ de Math., vol.~16, 130--132, 1851.

\Title{Notes on Monge's Applications}, see Monge.

\Author{Liouville, R.} \Title{Sur le caractère auquel se reconnaît l'équation
differentielle d'un système géodésique.} C.~R., vol.~108,
495--496, 1889.

\Title{Sur les représentations géodésiques des surfaces.} C.~R.,
vol.~108, 335--337, 1889.

\Author{Loria, G.} \Title{Sulla teoria della curvatura delle superficie.}
Rivista di Mat.\ Torino, vol.~2, 84--95, 1892.

\Title{Il passato ed il presente d.\ pr.\ Teorie geometriche.}
2nd~ed., 346~pp.\Add{,} Turin, 1896.

\Author{Lüroth, J.} \Title{Verallgemeinerung des Problems der kürzesten
Linien.} Zeitschrift für Math.\ und Phys., vol.~13, 156--160,
1868.

\Author{Mahler, E.} \Title{Ueber allgemeine Flächentheorie.} Archiv für
Math.\ and Phys., vol.~57, 96--97, 1881.

\Title{Die Fundamentalsätze der allgemeinen Flächentheorie.}
Vienna; Heft.~I, 1880; Heft.~II, 1881.

\Author{Mangeot, S.} \Title{Sur les éléments de~la courbure des courbes et
surfaces.} Ann.\ de l'École Normale, ser.~3, vol.~10, 87--89,
1893.

\Author{von Mangoldt, H.} \Title{Ueber diejenigen Punkte auf positiv
gekrümmten Flächen, welche die Eigenschaft haben,
dass die von ihnen ausgehenden geodätischen Linien nie
aufhören, kürzeste Linien zu sein.} Journ.\ für Math.,
vol.~91, 23--53, 1881.

\Title{Ueber die Klassification der Flächen nach der Verschiebbarkeit
ihrer geodätischen Dreiecke.} Journ.\ für
Math., vol.~94, 21--40, 1883.

\Author{Maxwell, J. Clerk.} \Title{On the Transformation of Surfaces by
Bending.} Trans.\ of Camb.\ Philos.\ Soc., vol.~9, 445--470,
1856.

\Author{Minding, Ferdinand.} \Title{Ueber die Biegung gewisser Flächen.}
Journ.\ für Math., vol.~18, 297--302, 365--368, 1838.

\Title{Wie sich entscheiden lässt, ob zwei gegebene krumme
Flächen auf einander abwickelbar sind oder nicht;
nebst Bemerkungen über die Flächen von veränderlichen
Krümmungsmaasse.} Journ.\ für Math., vol.~19,
370--387, 1839.

\Title{Beiträge zur Theorie der kürzesten Linien auf krummen
Flächen.} Journ.\ für Math., vol.~20, 323--327, 1840.

\Title{Ueber einen besondern Fall bei der Abwickelung
krummer Flächen.} Journ.\ für Math., vol.~20, 171--172,
1840.

\Title{Ueber die mittlere Krümmung der Flächen.} Bull.\
de l'Acad.\ Imp.\ de St.~Petersburg, vol.~20, 1875.

\Title{Zur Theorie der Curven kürzesten Umrings, bei
gegebenem Flächeninhalt, auf krummen Flächen.}
Journ.\ für Math., vol.~86, 279--289, 1879.

\Author{Mlodzieiowski, B.} \Title{Sur la déformation des surfaces.} Bull.\
de sci.\ Math., ser.~2, vol.~15, 97--101, 1891.

\Author{Monge, Gaspard.} \Title{Applications de l'Analyse à la Géométrie};
revue, corrigée et annotée par J.~Liouville. Paris;
fifth~ed., 1850.

\Author{Motoda, T.} Note to J.~Knoblauch's paper, ``\Title{Ueber Fundamentalgrössen
in der Flächentheorie}'' in Journ.\ für
Math., vol.~103. Journ.\ of the Phil.\ Soc.\ in Tokio,
3~pp., 1889.

\Author{Moutard, T. F.} \Title{Lignes de courbure d'une classe de surfaces
du quatrième ordre.} C.~R., vol.~59, 243, 1864.

\Title{Note sur la transformation par rayons vecteurs reciproques.}
Nouv.\ Ann.\ de Math.\ ser.~2, vol.~3, 306--309,
1864.

\Title{Sur les surface anallagmatique du quatrième ordre.}
Nouv.\ Ann.\ de Math.\ ser.~2, vol.~3, 536--539, 1864.

\Title{Sur la déformation des surfaces.} Bull.\ de~la Soc.\
Philomatique, p.~45, 1869.


\Title{Sur la construction des équations de~la forme $\dfrac{1}{x}·\Typo{\dfrac{d^{2}x}{dx\, dy}}{\dfrac{\dd^{2}x}{\dd x\, \dd y}}
= \lambda(x, y)$, qui admettent une intégrale générale explicite.}
Journ.\ de l'École Polyt., cahier~45, vol.~28, 1--11, 1878.

\Author{Nannei, E.} \Title{Le superficie ipercicliche.} Rend.\ dell'Accad.\
di Napoli, ser.~2, vol.~2, 119--121, 1888; Giornale di
Mat., vol.~26, 201--233, 1888.

\Author{Naccari, G.} \Title{Deduzioue delle principali formule relative
alla curvatura della superficie in generale e dello sferoide
in particolare con applicazione al meridiano di Venezia.}
L'Ateneo Veneto, ser.~17, vol.~1, 237--249, 1893; vol.~2,
133--161, 1893.

\Author{Padova, E.} \Title{Sopra un teorema di geometria differenziale.}
Reale Ist.\ Lombardo di sci.\ e let., vol.~23, 840--844, 1890.

\Title{Sulla teoria generale delle superficie.} Mem.\ della R.
Accad.\ dell' Ist.\ di Bologna, ser.~4, vol.~10, 745--772,
1890.

\Author{Pellet, A.} \Title{Mém.\ sur la théorie des surfaces et des courbes.}
Ann.\ de l'École Normale, ser.~3, vol.~14, 287--310, 1897.

\Title{Sur les surfaces de Weingarten.} C.~R., vol.~125, 601--602,
1897.

\Title{Sur les systèmes de surfaces orthogonales et isothermes.}
C.~R., vol.~124, 552--554, 1897.

\Title{Sur les surfaces ayant même représentation sphérique.}
C.~R., vol.~124, 1291--1294, 1897.

\Title{Sur les surfaces \Typo{isometriques}{isométriques}.} C.~R., vol.~124, 1337--1339,
1897.

\Title{Sur la théorie des surfaces.} Bull.\ de~la Soc.\ Math.\ de
France, vol.~26, 138--159, 1898; C.~R., vol.~124, 451--452,
739--741, 1897; Thesis, Paris, 1878.

\Title{Sur les surfaces applicables sur une surface de \Typo{revolution}{révolution}.}
C.~R., vol.~125, 1159--1160, 1897; vol.~126, 392--394,
1898.

\Author{Peter, A.} \Title{Die Flächen, deren Haupttangentencurven linearen
Complexen angehören.} Archiv for Math.\ og
Nat., Christiania, vol.~17, No.~8, 1--91, 1895.
\PageSep{124}

\Author{Petot, A.} \Title{Sur les surfaces dont l'élément \Typo{lineaire}{linéaire} est \Typo{reductible}{réductible}
\Typo{a}{à} la forme $ds^{2} = F(U + V)(du^{2} + dv^{2})$.} C.~R.,
vol.~110, 330--333, 1890.

\Author{Picard, Émile.} \Title{Surfaces applicables.} Traité d'Analyse,
vol.~1, chap.~15, 420--457; first~ed., 1891; second~ed.,
1901.

\Author{Pirondini, G.} \Title{Studi geometrici relativi specialmente alle
superficie gobbe.} Giornale di Mat., vol.~23, 288--331,
1885.

\Title{Teorema relativo alle linee di curvatura delle superficie
e sue applicazioni.} Annali di Mat., ser.~2, vol.~16,
61--84, 1888; vol.~21, 33--46, 1893.

\Author{Plücker, Julius.} \Title{Ueber die Krümmung einer beliebigen
Fläche in einem gegebenen Puncte.} Journ.\ für Math.,
vol.~3, 324--336, 1828.

\Author{Poincaré, H.} \Title{Rapport sur un Mémoire de M.~Hadamard,
intitulé: Sur les lignes géodésiques des surfaces à courbures
opposées.} C.~R., vol.~125, 589--591, 1897.

\Author{Probst, F.} \Title{Ueber Flächen mit isogonalen systemen von
geodätischen Kreisen.} Inaug.-diss.\ 46~pp., Würzburg,
1893.

\Author{Raffy, L.} \Title{Sur certaines surfaces, dont les rayons de courbure
sont liés par une relation.} Bull.\ de~la Soc.\ Math.\
de France, vol.~19, 158--169, 1891.

\Title{\Typo{Determination}{Détermination} des éléments linéaires doublement harmoniques.}
Journ.\ de Math., ser.~4, vol.~10, 331--390,
1894.

\Title{Quelques \Typo{proprietes}{propriétés} des surfaces harmoniques.} Ann.\
de~la Faculté des sci.\ de Toulouse, vol.~9~C, 1--44, 1895.

\Title{Sur les spirales harmoniques.} Ann.\ de l'École Normale,
ser.~3, vol.~12, 145--196, 1895.

\Title{Surfaces rapportées à un réseau conjugué azimutal.}
Bull.\ de~la Soc.\ Math.\ de France, vol.~24, 51--56, 1896.

\Title{Leçons sur les applications géométriques de l'analyse.}
Paris, \textsc{vi}~+~251~pp., 1897.

\Title{Contribution à la théorie des surfaces dont les rayons
de courbure sont liés par une relation.} Bull.\ de~la Soc.\
Math.\ de France, vol.~25, 147--172, 1897.

\Title{Sur les formules fondamentales de~la théorie des surfaces.}
Bull.\ de~la Soc.\ Math.\ de France, vol.~25, 1--3,
1897.

\Title{Détermination d'une surface par ses deux formes quadratiques
fondamentales.} C.~R., vol.~126, 1852--1854,
1898.

\Author{\Typo{Razziboni}{Razzaboni}, Amilcare.} \Title{Sulla rappresentazzione di una superficie
su di un' altra al modo di Gauss.} Giornali di Mat.,
vol.~27, 274--302, 1889.

\Title{Delle superficie sulle quali due serie di geodetiche
formano un sistema conjugato.} Mem.\ della R. Accad.\
dell'Ist.\ di Bologna, ser.~4, vol.~9, 765--776, 1889.

\Author{Reina, V.} \Title{Sulle linee conjugate di una superficie.} Atti
della Reale Accad.\ dei Lincei, ser.~4, vol.~$6_{1}$, 156--165,
203--209, 1890.

%\Author{Reina, V.}
\Title{Di alcune formale relative alla teoria delle superficie.}
Atti della Reale Accad.\ dei Lincei, ser.~4, vol.~$6_{2}$\Add{,}
103--110, 176, 1890.

\Author{Resal, H.} \Title{Exposition de~la théorie des surfaces.} 1~vol.,
\textsc{xiii}~+~171~pp.\Add{,} Paris, 1891. Bull.\ des sci.\ Math., ser.~2,
vol.~15, 226--227, 1891; Journ.\ de Math.\ spèciale à
l'usage des candidats aux École Polyt., ser.~3, vol.~5,
165--166, 1891.

\Author{Ribaucour, A.} \Title{Sur la théorie de l'application des surfaces
l'une sur l'autre.} L'Inst.\ Journ.\ universel des sci.\ et
des soc.\ sav.\ en France, sect.~I, vol.~37, 371--382, 1869.

\Title{Sur les surfaces orthogonales.} L'Inst. Journ.\ universel
des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~37,
29--30, 1869.

\Title{Sur la déformation des surfaces.} L'Inst. Journ.\ universel
des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~37,
389, 1869; C.~R., vol.~70, 330, 1870.

\Title{Sur la théorie des surfaces.} L'Inst.\ Journ.\ universel
des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~38, 60--61,
141--142, 236--237, 1870.

\Title{Sur la représentation sphérique des surfaces.} C.~R.,
vol.~75, 533--536, 1872.

\Title{Sur les courbes enveloppes de cercles et sur les surfaces
enveloppes de sphères.} Nouvelle Correspondance
Math., vol.~5, 257--263, 305--315, 337--343, 385--393, 417--425,
1879; vol.~6, 1--7, 1880.

\Title{Mémoire sur la théorie générale des surfaces courbes.}
Journ.\ de Math., ser.~4, vol.~7, 5--108, 219--270, 1891.

\Author{Ricci, G.} \Title{Dei sistemi di coordinate atti a ridurre la expressione
del quadrato dell' elemento lineaire di una superficie
alla forma $ds^{2} = (U + V)(du^{2} + dv^{2})$.} Atti della
Reale Accad.\ dei Lincei, ser.~5, vol.~$2_{1}$, 73--81, 1893.

\Title{A proposito di una memoria sulle linee geodetiche del
sig. G.~Königs.} Atti della Reale Accad.\ dei Lincei, ser.~5,
vol.~$2_{2}$, 146--148, 338--339, 1893.

\Title{Sulla teoria delle linee geodetiche e dei sistemi isotermi
di Liouville.} Atti del Reale Ist.\ Veneto, ser.~7, vol.~5,
643--681, 1894.

\Title{Della equazione fondamentale di Weingarten nella
teoria delle superficie applicabili.} Atti del Reale Inst.\
Veneto, ser.~7, vol.~8, 1230--1238, 1897.

\Title{Lezioni sulla teoria delle superficie.} \textsc{viii}~+~416~pp.\Add{,}
Verona, 1898.

\Author{Rothe, R.} \Title{Untersuchung über die Theorie der isothermen
Flächen.} Diss., 42~pp.\Add{,} Berlin, 1897.

\Author{Röthig, O.} \Title{Zur Theorie der Flächen.} \Typo{Jouru.}{Journ.}\ für Math.,
vol.~85, 250--263, 1878.

\Author{Ruffini, F.} \Title{Di alcune proprietà della rappresentazione
sferica del Gauss.} Mem.\ dell'Accad.\ Reale di sci.\
dell'Ist.\ di Bologna, ser.~4, vol.~8, 661--680, 1887.

\Author{Ruoss, H.} \Title{Zur Theorie des Gauss'schen Krümmungsmaases.}
Zeitschrift für Math.\ und Phys., vol.~37, 378--381,
1892.
\PageSep{125}

\Author{Saint Loup.} \Title{Sur les propriétés des lignes géodésiques.}
Thesis, 33--96, Paris, 1857.

\Author{Salmon, George.} \Title{Analytische Geometrie des Raumes.} Revised
by Wilhelm Fielder. Vol.~II, \textsc{lxxii}~+~696\Chg{;}{~pp.,}
Leipzig, 1880.

\Author{de Salvert, F.} \Title{Mémoire sur la théorie de~la courbure des
surfaces.} Ann.\ de~la Soc.\ sci.\ de Bruxelles, vol.~5~B,
291--473, 1881; Paris, Gauthier-villars, 1881.

\Title{Mémoire sur l'emploi des coordonnées curvilignes
dans les problèmes de Mècanique et les lignes géodésiques
des surfaces isothermes.} Ann.\ de~la Soc.\ sci.\ de Bruxelles,
vol.~11~B, 1--138, 1887. Paris, 1887.

\Title{Mémoire sur la recherche la plus générale d'un système
orthogonal triplement isotherme.} Ann.\ de~la Soc.\
sci.\ de Bruxelles, vol.~13~B, 117--260, 1889; vol.~14~B,
121--283, 1890; vol.~15~B, 201--394, 1891; vol.~16~B,
273--366, 1892; vol.~17~B, 103--272, 1893; vol.~18~B, 61--64,
1894.

\Title{Théorie nouvelle du système orthogonal triplement
isotherme et son application aux coordonnées curvilignes.}
2~vols., Paris, 1894.

\Author{Scheffers, G.} \Title{Anwendung der Differential- und Integralrechung
auf Geometrie.} vol.~I, \textsc{x}~+~360~pp., Leipzig,
Veit~\&~Co., 1901.

\Author{Schering, E.} \Title{Erweiterung des Gauss'schen Fundamentalsatzes
für Dreiecke in stetig gekrümmten Flächen.}
Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
1867, 389--391; 1868, 389--391.

\Author{Serret, Paul.} \Title{Sur la courbure des surfaces.} C.~R., vol.~84,
543--546, 1877.

\Author{Servais, C.} \Title{Sur la courbure dans les surfaces.} Bull.\ de
l'Acad.\ Royale de Belgique, ser.~3, vol.~24, 467--474,
1892.

\Title{Quelques formules sur la courbure des surfaces.} Bull.\
de l'Acad.\ Royale de Belgique, ser.~3, vol.~27, 896--904,
1894.

\Author{Simonides, J.} \Title{Ueber die Krümmung der Flächen.} Zeitschrift
zur Pflege der Math.\ und Phys., vol.~9, 267, 1880.

\Author{Stäckel, Paul.} \Title{Zur Theorie des Gauss'schen Krümmungsmaasses.}
Journ.\ für Math., vol.~111, 205--206, 1893;
Berichte der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
vol.~45, 163--169, 170--172, 1893.

\Title{Bemerkungen zur Geschichte der geodätischen Linien.}
Berichte der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
vol.~45, 444--467, 1893.

\Title{Sur la déformation des surfaces.} C.~R., vol.~123, 677--680,
1896.

\Title{Biegungen und conjugirte Systeme.} Math.\ Annalen,
vol.~49, 255--310, 1897.

\Title{Beiträge zur Flächentheorie.} Berichte der Kgl.\ Gesell.\
der Wissenschaften zu Leipzig, vol.~48, 478--504, 1896;
vol.~50, 3--20, 1898.

\Author{Stahl und Kommerell.} \Title{Die Grundformeln der allgemeinen
Flächentheorie.} \textsc{vi}~+~114~pp., Leipzig, 1893.

\Author{Staude, O.} \Title{Ueber das Vorzeichen der geodätischen Krümmung.}
Dorpat Naturf.\ Ges.\ Ber., 1895, 72--83.

\Author{Stecker, H. F.} \Title{On the determination of surfaces capable of
conformal representation upon the plane in such a manner
that geodetic lines are represented by algebraic
curves.} Trans.\ Amer.\ Math.\ Society, vol.~2, 152--165,
1901.

\Author{Stouff, X.} \Title{Sur la valeur de la courbure totale d'une surface
aux points d'une arête de rebroussement.} Ann.\ de
l'École Normale, ser.~3, vol.~9, 91--100, 1892.

\Author{Sturm, Rudolf.} \Title{Ein Analogon zu Gauss' Satz von der Krümmung
der Flächen.} Math.\ Annalen, vol.~21, 379--384,
1883.

\Author{Stuyvaert, M.} \Title{Sur la courbure des lignes et des surfaces\Add{.}}
Mém.\ couronnés et autr.\ mém.\ publ.\ par l'Acad.\ Royale
de Belgique, vol.~55, 19 pp., 1898.

\Author{de Tannenberg, W.} \Title{Leçons sur les applications géométriques
du calcul differentiel.} 192~pp.\Add{,} Paris, A.~Hermann,
1899.

\Author{van Dorsten}, see (van) Dorsten.

\Author{von Escherich}, see (von) Escherich.

\Author{von Lilienthal}, see (von) Lilienthal.

\Author{von Mangoldt}, see (von) Mangoldt.

\Author{Vivanti, G.} \Title{Ueber diejenigen Berührungstransformationen,
welche das Verhältniss der Krümmungsmaasse irgend
zwei sich berührender Flächen im Berührungspunkte
unverändert lassen.} Zeitschrift für Math.\ und Phys.,
vol.~37, 1--7, 1892.

\Title{Sulle superficie a curvatura media costante.} Reale
Ist.\ Lombardo di sci.\ e let.\ Milano. Ser.~2, vol.~28, 353--364,
1895.

\Author{Voss, A.} \Title{Ueber ein neues Princip der Abbildung krummer
Oberflächen.} Math.\ Annalen, vol.~19, 1--26, 1882.

\Title{Ueber dirjenigen Flächen, auf denen zwei Scharen
geodätischer Linien ein conjugirtes System bilden.}
Sitzungsb.\ der Kgl.\ Bayer.\ Akad.\ der Wissenschaften
zu München, vol.~18, 95--102, 1888.

\Title{Zur Theorie der Krümmung der Flächen.} Math.\
Annalen, vol.~39, 179--256, 1891.

\Title{Ueber die Fundamentalgleichungen der Flächentheorie.}
\Typo{Sitzungb.}{Sitzungsb.}\ der Kgl.\ Bayer\Add{.} Akad.\ der Wissenschaften
zu München, vol.~22, 247--278, 1892.

\Title{Ueber isometrische Flächen.} Math.\ Annalen, vol.~46,
97--132, 1895.

\Title{Ueber infinitesimale Flächendeformationen.} Jahresb.\
der Deutschen Math.\ Vereinigung, vol.~4, 132--137,
1897.

\Title{Zur Theorie der infinitesimalen Biegungsdeformationen
einer Fläche.} Sitzungsb.\ der Kgl.\ Akad.\ der Wissenschaften
zu München, vol.~27, 229--301, 1897.
\PageSep{126}

\Author{Waelsch, E.} \Title{Sur les surfaces à élément linéaire de Liouville
et les surfaces à courbure constante.} C.~R., vol.~116,
1435--1437, 1893.

\Title{Sur les lignes géodésiques de certaines surfaces.} C.~R.,
vol.~125, 521--523, 1897.

\Title{Ueber Flächen mit \Typo{Liouville'schem}{Liouville'schen} Bogenelement.}
Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften zu Wien,
vol.~106, 323--328, 1897.

\Author{Warren, J. W.} \Title{An improved form of writing the formula
of C.~F.~Gauss for the measure of curvature.} Quart.\
Journ.\ of Math., vol.~16, 219--224, 1879.

\Title{Exercises in curvilinear and normal coordinates.}
Trans.\ of Camb.\ Philos.\ Society, vol.~12, 455--522, 531--545,
1879.

\Author{Weierstrass, Karl.} \Title{Ueber die Flächen, deren mittlere
Krümmung überall gleich Null ist.} Monatsb.\ der
Akad.\ der Wissenschaften zu Berlin, 1866, 612--625.

\Author{Weingarten, Julius.} \Title{Ueber eine Klasse auf einander abwickelbarer
Flächen.} Journ.\ für Math., vol.~59, 382--393,
1861.

\Title{Ueber die Flächen deren Normalen eine gegebene
Fläche berühren.} Journ.\ für Math., vol.~62, 61--63, 1863.

\Title{Ueber die Oberflächen, für welche einer der beiden
Hauptkrümmungshalbmesser eine Function des andern
ist.} Journ.\ für Math., vol.~62, 160--173, 1863; vol.~103,
184, 1888.

\Title{Ueber die Verschiebbarkeit geodätischer Dreiecke in
krummen Flächen.} Sitzungsb.\ der Kgl.\ Akad.\ der
Wissenschaften zu Berlin, 1882, 453--456.

\Title{Ueber die Eigenschaften des Linienelements der
Flächen von constantem Krümmungsmaass.} Journ.\ für
Math., vol.~94, 181--202, 1883; vol.~95, 325--329, 1883.

\Title{Ueber die Theorie der auf einander Abwickelbaren
Oberflächen.} Festschrift d.\ Techn.\ Hochschule Berlin,
1884.

\Title{Ueber die unendlich kleinen Deformationen einer
biegsamen, unausdehnbahren Fläche.} Sitzungsb.\ der
Kgl.\ Akad.\ der Wissenschaften zu Berlin, 1886, 83--91.

\Title{Eine neue Klasse auf einander abwickelbarer Flächen.}
Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
1887, 28--31.

%\Author{Weingarten, Julius.}
\Title{Ueber die Deformationen einer biegsamen
unausdehnbaren Fläche.} Journ.\ für Math., vol.~100,
296--310, 1887.

\Title{Sur la théorie des surfaces applicables sur une surface
donnée.} Extrait d'une lettre à M.~Darboux. C.~R.,
vol.~112, 607--610, 706--707, 1891.

\Title{Sur la déformation des surfaces.} Acta Mathematica,
vol.~20, 159--200, 1897; note on same, vol.~22, 193--199,
1899.

\Author{Weyr, Ed.} \Title{Sur l'équation des lignes géodésiques.} Chicago
Congr.\ Papers, 408--411, 1896.

\Title{Ueber das System der Orthogonalflächen.} Zeitschrift
zur Pflege der Math.\ und Phys., vol.~25, 42--46, 1896.

\Author{Willgrod, Heinrich.} \Title{Ueber Flächen, welche sich durch
ihre Krümmungslinien in unendlich kleine Quadrate
theilen lassen.} Diss., \textsc{vi}~+~51~pp., Göttingen, 1883.

\Author{Williamson, Benjamin.} \Title{On curvilinear coordinates.} Trans.\
of the Royal Irish Acad., Dublin, vol.~29, part~15, 515--552,
1890.

\Title{On Gauss's theorem of the measure of curvature at
any point of a surface.} Quart.\ Journ.\ of Math., vol.~11,
362--366, 1871.

\Author{Wostokow, J.} \Title{On the geodesic curvature of curves on a
surface}, (Russian.) Works of the Warsaw Soc.\ of Sci.,
sect.~6, No.~8, 1896.

\Author{Woudstra, M.} \Title{Kromming van oppervlakken volgens de
theorie van Gauss.} Diss., Groningen, 1879.

\Author{Zorawski, K.} \Title{On deformation of surfaces.} Trans.\ of the
Krakau Acad.\ of Sciences, (Polish), ser.~2, vol.~1, 225--291,
1891.

\Title{Ueber Biegungsinvarianten. Eine Anwendung der
Lie'schen Gruppentheorie.} Diss., Leipzig, 1891; Acta
Mathematica, vol.~16, 1--64, 1892.

\Title{On the fundamental magnitudes of the general theory
of surfaces.} Memoirs of the Krakau Acad.\ of Science,
(Polish), vol.~28, 1--7, 1895.

\Title{On some relations in the theory of surfaces.} Bull.\ of
the Krakau Acad.\ of Sciences, (Polish), vol.~33, 106--119,
1898.
\end{Bibliography}
\PageSep{127}

\iffalse
CORRIGENDA BT ADDENDA.
%[** TN: [x] = corrected in source using \Erratum macro,
%        [v] = verified in source, corrected by the translators]

[x] Art. 11, p. 20, l. 6. The fourth E should be F.

[x] Art. 18, p. 27, l. 7. For \sqrt{EG - F^2)·dp·d\theta read 2\sqrt{FG - F^w)·dq·d\theta.
The original and the Latin reprints lack the factor 2; the correction is made in all
the translations.

[x] Art. 19, p. 28, l. 10. For g read q.

[v] Art. 22, p. 34, l. 5, left side; Art. 24, p. 36, l. 5, third equation; Art. 24,
p. 38, l. 4. The original and Liouville's reprint have q for p.

[x] Note on Art. 23, p. 55, l. 2 fr. bot. For p read q.
\fi

%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%%
\PGLicense
\begin{PGtext}
End of the Project Gutenberg EBook of General Investigations of Curved
Surfaces of 1827 and 1825, by Karl Friedrich Gauss

*** END OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***

***** This file should be named 36856-tex.tex or 36856-tex.zip *****
This and all associated files of various formats will be found in:
        http://www.gutenberg.org/3/6/8/5/36856/

Produced by Andrew D. Hwang, with special thanks to Brenda Lewis.

Updated editions will replace the previous one--the old editions
will be renamed.

Creating the works from public domain print editions means that no
one owns a United States copyright in these works, so the Foundation
(and you!) can copy and distribute it in the United States without
permission and without paying copyright royalties.  Special rules,
set forth in the General Terms of Use part of this license, apply to
copying and distributing Project Gutenberg-tm electronic works to
protect the PROJECT GUTENBERG-tm concept and trademark.  Project
Gutenberg is a registered trademark, and may not be used if you
charge for the eBooks, unless you receive specific permission.  If you
do not charge anything for copies of this eBook, complying with the
rules is very easy.  You may use this eBook for nearly any purpose
such as creation of derivative works, reports, performances and
research.  They may be modified and printed and given away--you may do
practically ANYTHING with public domain eBooks.  Redistribution is
subject to the trademark license, especially commercial
redistribution.



*** START: FULL LICENSE ***

THE FULL PROJECT GUTENBERG LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg-tm mission of promoting the free
distribution of electronic works, by using or distributing this work
(or any other work associated in any way with the phrase "Project
Gutenberg"), you agree to comply with all the terms of the Full Project
Gutenberg-tm License (available with this file or online at
http://gutenberg.net/license).


Section 1.  General Terms of Use and Redistributing Project Gutenberg-tm
electronic works

1.A.  By reading or using any part of this Project Gutenberg-tm
electronic work, you indicate that you have read, understand, agree to
and accept all the terms of this license and intellectual property
(trademark/copyright) agreement.  If you do not agree to abide by all
the terms of this agreement, you must cease using and return or destroy
all copies of Project Gutenberg-tm electronic works in your possession.
If you paid a fee for obtaining a copy of or access to a Project
Gutenberg-tm electronic work and you do not agree to be bound by the
terms of this agreement, you may obtain a refund from the person or
entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B.  "Project Gutenberg" is a registered trademark.  It may only be
used on or associated in any way with an electronic work by people who
agree to be bound by the terms of this agreement.  There are a few
things that you can do with most Project Gutenberg-tm electronic works
even without complying with the full terms of this agreement.  See
paragraph 1.C below.  There are a lot of things you can do with Project
Gutenberg-tm electronic works if you follow the terms of this agreement
and help preserve free future access to Project Gutenberg-tm electronic
works.  See paragraph 1.E below.

1.C.  The Project Gutenberg Literary Archive Foundation ("the Foundation"
or PGLAF), owns a compilation copyright in the collection of Project
Gutenberg-tm electronic works.  Nearly all the individual works in the
collection are in the public domain in the United States.  If an
individual work is in the public domain in the United States and you are
located in the United States, we do not claim a right to prevent you from
copying, distributing, performing, displaying or creating derivative
works based on the work as long as all references to Project Gutenberg
are removed.  Of course, we hope that you will support the Project
Gutenberg-tm mission of promoting free access to electronic works by
freely sharing Project Gutenberg-tm works in compliance with the terms of
this agreement for keeping the Project Gutenberg-tm name associated with
the work.  You can easily comply with the terms of this agreement by
keeping this work in the same format with its attached full Project
Gutenberg-tm License when you share it without charge with others.

1.D.  The copyright laws of the place where you are located also govern
what you can do with this work.  Copyright laws in most countries are in
a constant state of change.  If you are outside the United States, check
the laws of your country in addition to the terms of this agreement
before downloading, copying, displaying, performing, distributing or
creating derivative works based on this work or any other Project
Gutenberg-tm work.  The Foundation makes no representations concerning
the copyright status of any work in any country outside the United
States.

1.E.  Unless you have removed all references to Project Gutenberg:

1.E.1.  The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg-tm License must appear prominently
whenever any copy of a Project Gutenberg-tm work (any work on which the
phrase "Project Gutenberg" appears, or with which the phrase "Project
Gutenberg" is associated) is accessed, displayed, performed, viewed,
copied or distributed:

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net

1.E.2.  If an individual Project Gutenberg-tm electronic work is derived
from the public domain (does not contain a notice indicating that it is
posted with permission of the copyright holder), the work can be copied
and distributed to anyone in the United States without paying any fees
or charges.  If you are redistributing or providing access to a work
with the phrase "Project Gutenberg" associated with or appearing on the
work, you must comply either with the requirements of paragraphs 1.E.1
through 1.E.7 or obtain permission for the use of the work and the
Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or
1.E.9.

1.E.3.  If an individual Project Gutenberg-tm electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs 1.E.1 through 1.E.7 and any additional
terms imposed by the copyright holder.  Additional terms will be linked
to the Project Gutenberg-tm License for all works posted with the
permission of the copyright holder found at the beginning of this work.

1.E.4.  Do not unlink or detach or remove the full Project Gutenberg-tm
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg-tm.

1.E.5.  Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1.E.1 with
active links or immediate access to the full terms of the Project
Gutenberg-tm License.

1.E.6.  You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including any
word processing or hypertext form.  However, if you provide access to or
distribute copies of a Project Gutenberg-tm work in a format other than
"Plain Vanilla ASCII" or other format used in the official version
posted on the official Project Gutenberg-tm web site (www.gutenberg.net),
you must, at no additional cost, fee or expense to the user, provide a
copy, a means of exporting a copy, or a means of obtaining a copy upon
request, of the work in its original "Plain Vanilla ASCII" or other
form.  Any alternate format must include the full Project Gutenberg-tm
License as specified in paragraph 1.E.1.

1.E.7.  Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg-tm works
unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8.  You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg-tm electronic works provided
that

- You pay a royalty fee of 20% of the gross profits you derive from
     the use of Project Gutenberg-tm works calculated using the method
     you already use to calculate your applicable taxes.  The fee is
     owed to the owner of the Project Gutenberg-tm trademark, but he
     has agreed to donate royalties under this paragraph to the
     Project Gutenberg Literary Archive Foundation.  Royalty payments
     must be paid within 60 days following each date on which you
     prepare (or are legally required to prepare) your periodic tax
     returns.  Royalty payments should be clearly marked as such and
     sent to the Project Gutenberg Literary Archive Foundation at the
     address specified in Section 4, "Information about donations to
     the Project Gutenberg Literary Archive Foundation."

- You provide a full refund of any money paid by a user who notifies
     you in writing (or by e-mail) within 30 days of receipt that s/he
     does not agree to the terms of the full Project Gutenberg-tm
     License.  You must require such a user to return or
     destroy all copies of the works possessed in a physical medium
     and discontinue all use of and all access to other copies of
     Project Gutenberg-tm works.

- You provide, in accordance with paragraph 1.F.3, a full refund of any
     money paid for a work or a replacement copy, if a defect in the
     electronic work is discovered and reported to you within 90 days
     of receipt of the work.

- You comply with all other terms of this agreement for free
     distribution of Project Gutenberg-tm works.

1.E.9.  If you wish to charge a fee or distribute a Project Gutenberg-tm
electronic work or group of works on different terms than are set
forth in this agreement, you must obtain permission in writing from
both the Project Gutenberg Literary Archive Foundation and Michael
Hart, the owner of the Project Gutenberg-tm trademark.  Contact the
Foundation as set forth in Section 3 below.

1.F.

1.F.1.  Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread
public domain works in creating the Project Gutenberg-tm
collection.  Despite these efforts, Project Gutenberg-tm electronic
works, and the medium on which they may be stored, may contain
"Defects," such as, but not limited to, incomplete, inaccurate or
corrupt data, transcription errors, a copyright or other intellectual
property infringement, a defective or damaged disk or other medium, a
computer virus, or computer codes that damage or cannot be read by
your equipment.

1.F.2.  LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
of Replacement or Refund" described in paragraph 1.F.3, the Project
Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg-tm trademark, and any other party distributing a Project
Gutenberg-tm electronic work under this agreement, disclaim all
liability to you for damages, costs and expenses, including legal
fees.  YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH 1.F.3.  YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.

1.F.3.  LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
defect in this electronic work within 90 days of receiving it, you can
receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from.  If you
received the work on a physical medium, you must return the medium with
your written explanation.  The person or entity that provided you with
the defective work may elect to provide a replacement copy in lieu of a
refund.  If you received the work electronically, the person or entity
providing it to you may choose to give you a second opportunity to
receive the work electronically in lieu of a refund.  If the second copy
is also defective, you may demand a refund in writing without further
opportunities to fix the problem.

1.F.4.  Except for the limited right of replacement or refund set forth
in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER
WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.

1.F.5.  Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of damages.
If any disclaimer or limitation set forth in this agreement violates the
law of the state applicable to this agreement, the agreement shall be
interpreted to make the maximum disclaimer or limitation permitted by
the applicable state law.  The invalidity or unenforceability of any
provision of this agreement shall not void the remaining provisions.

1.F.6.  INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg-tm electronic works in accordance
with this agreement, and any volunteers associated with the production,
promotion and distribution of Project Gutenberg-tm electronic works,
harmless from all liability, costs and expenses, including legal fees,
that arise directly or indirectly from any of the following which you do
or cause to occur: (a) distribution of this or any Project Gutenberg-tm
work, (b) alteration, modification, or additions or deletions to any
Project Gutenberg-tm work, and (c) any Defect you cause.


Section  2.  Information about the Mission of Project Gutenberg-tm

Project Gutenberg-tm is synonymous with the free distribution of
electronic works in formats readable by the widest variety of computers
including obsolete, old, middle-aged and new computers.  It exists
because of the efforts of hundreds of volunteers and donations from
people in all walks of life.

Volunteers and financial support to provide volunteers with the
assistance they need are critical to reaching Project Gutenberg-tm's
goals and ensuring that the Project Gutenberg-tm collection will
remain freely available for generations to come.  In 2001, the Project
Gutenberg Literary Archive Foundation was created to provide a secure
and permanent future for Project Gutenberg-tm and future generations.
To learn more about the Project Gutenberg Literary Archive Foundation
and how your efforts and donations can help, see Sections 3 and 4
and the Foundation web page at http://www.pglaf.org.


Section 3.  Information about the Project Gutenberg Literary Archive
Foundation

The Project Gutenberg Literary Archive Foundation is a non profit
501(c)(3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service.  The Foundation's EIN or federal tax identification
number is 64-6221541.  Its 501(c)(3) letter is posted at
http://pglaf.org/fundraising.  Contributions to the Project Gutenberg
Literary Archive Foundation are tax deductible to the full extent
permitted by U.S. federal laws and your state's laws.

The Foundation's principal office is located at 4557 Melan Dr. S.
Fairbanks, AK, 99712., but its volunteers and employees are scattered
throughout numerous locations.  Its business office is located at
809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email
business@pglaf.org.  Email contact links and up to date contact
information can be found at the Foundation's web site and official
page at http://pglaf.org

For additional contact information:
     Dr. Gregory B. Newby
     Chief Executive and Director
     gbnewby@pglaf.org


Section 4.  Information about Donations to the Project Gutenberg
Literary Archive Foundation

Project Gutenberg-tm depends upon and cannot survive without wide
spread public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine readable form accessible by the widest
array of equipment including outdated equipment.  Many small donations
($1 to $5,000) are particularly important to maintaining tax exempt
status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States.  Compliance requirements are not uniform and it takes a
considerable effort, much paperwork and many fees to meet and keep up
with these requirements.  We do not solicit donations in locations
where we have not received written confirmation of compliance.  To
SEND DONATIONS or determine the status of compliance for any
particular state visit http://pglaf.org

While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.

International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States.  U.S. laws alone swamp our small staff.

Please check the Project Gutenberg Web pages for current donation
methods and addresses.  Donations are accepted in a number of other
ways including including checks, online payments and credit card
donations.  To donate, please visit: http://pglaf.org/donate


Section 5.  General Information About Project Gutenberg-tm electronic
works.

Professor Michael S. Hart is the originator of the Project Gutenberg-tm
concept of a library of electronic works that could be freely shared
with anyone.  For thirty years, he produced and distributed Project
Gutenberg-tm eBooks with only a loose network of volunteer support.


Project Gutenberg-tm eBooks are often created from several printed
editions, all of which are confirmed as Public Domain in the U.S.
unless a copyright notice is included.  Thus, we do not necessarily
keep eBooks in compliance with any particular paper edition.


Most people start at our Web site which has the main PG search facility:

     http://www.gutenberg.net

This Web site includes information about Project Gutenberg-tm,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.
\end{PGtext}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% End of the Project Gutenberg EBook of General Investigations of Curved  %
% Surfaces of 1827 and 1825, by Karl Friedrich Gauss                      %
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
%                                                                         %
% ***** This file should be named 36856-tex.tex or 36856-tex.zip *****    %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/3/6/8/5/36856/                         %
%                                                                         %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

\end{document}
###
@ControlwordReplace = (
  ['\\Introduction', 'Introduction'],
  ['\\Contents', 'Contents'],
  ['\\Notes', 'Notes'],
  ['\\BibliographyPage', 'Bibliography'],
  ['\\end{Theorem}', ''],
  ['\\end{Bibliography}', ''],
  ['\\tb', '-----'],
  ['\\ie', 'i.e.'],
  ['\\eg', 'e.g.'],
  ['\\Cf', 'Cf.'],
  ['\\QEA', 'Q.E.A.']
  );

@ControlwordArguments = (
  ['\\setlength', 1, 0, '', '', 1, 0, '', ''],
  ['\\SetRunningHeads', 1, 0, '', '', 1, 0, '', ''],
  ['\\BookMark', 1, 0, '', '', 1, 0, '', ''],
  ['\\Article', 1, 1, '', ''],
  ['\\Par', 1, 1, '', ''],
  ['\\Abstract', 1, 1, '', ''],
  ['\\ToCLine', 1, 1, '', '', 1, 0, '', ''],
  ['\\Signature', 1, 1, '', ' ', 1, 1, '', ''],
  ['\\Publication', 1, 1, '', ' ', 1, 1, '', ' ', 1, 1, '', ''],
  ['\\Author', 1, 1, '', ''],
  ['\\Title', 1, 1, '', ''],
  ['\\LineRef', 0, 0, '', '', 1, 0, '', '', 1, 1, '', ''],
  ['\\LineRefs', 1, 1, '', ''],
  ['\\Art', 1, 1, 'Art. ', ''],
  ['\\Pageref', 0, 0, '', '', 1, 1, 'page ', ''],
  ['\\begin{Theorem}', 0, 0, '', ''],
  ['\\begin{Bibliography}', 1, 1, '', ''],
  ['\\Note', 1, 0, '', ''],
  ['\\Typo', 1, 0, '', '', 1, 1, '', ''],
  ['\\Add', 1, 1, '', ''],
  ['\\Chg', 1, 0, '', '', 1, 1, '', ''],
  ['\\PageSep', 1, 1, '%%-- Page [', ']'],
  ['\\Figure', 0, 0, '', '', 1, 0, '', ''],
  ['\\Graphic', 1, 0, '', '', 1, 0, '', '']
  );
###
This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2010.5.6)  25 JUL 2011 16:26
entering extended mode
 %&-line parsing enabled.
**36856-t.tex
(./36856-t.tex
LaTeX2e <2005/12/01>
Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
n, swedish, ukenglish, pinyin, loaded.
(/usr/share/texmf-texlive/tex/latex/base/book.cls
Document Class: book 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/leqno.clo
File: leqno.clo 1998/08/17 v1.1c Standard LaTeX option (left equation numbers)
) (/usr/share/texmf-texlive/tex/latex/base/bk12.clo
File: bk12.clo 2005/09/16 v1.4f Standard LaTeX file (size option)
)
\c@part=\count79
\c@chapter=\count80
\c@section=\count81
\c@subsection=\count82
\c@subsubsection=\count83
\c@paragraph=\count84
\c@subparagraph=\count85
\c@figure=\count86
\c@table=\count87
\abovecaptionskip=\skip41
\belowcaptionskip=\skip42
\bibindent=\dimen102
) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
Package: inputenc 2006/05/05 v1.1b Input encoding file
\inpenc@prehook=\toks14
\inpenc@posthook=\toks15
(/usr/share/texmf-texlive/tex/latex/base/latin1.def
File: latin1.def 2006/05/05 v1.1b Input encoding file
)) (/usr/share/texmf-texlive/tex/latex/base/fontenc.sty
Package: fontenc 2005/09/27 v1.99g Standard LaTeX package
(/usr/share/texmf-texlive/tex/latex/base/t1enc.def
File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file
LaTeX Font Info:    Redeclaring font encoding T1 on input line 43.
)) (/usr/share/texmf-texlive/tex/latex/base/ifthen.sty
Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
Package: amsmath 2000/07/18 v2.13 AMS math features
\@mathmargin=\skip43
For additional information on amsmath, use the `?' option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
Package: amstext 2000/06/29 v2.01
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty
File: amsgen.sty 1999/11/30 v2.0
\@emptytoks=\toks16
\ex@=\dimen103
)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty
Package: amsbsy 1999/11/29 v1.2d
\pmbraise@=\dimen104
) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty
Package: amsopn 1999/12/14 v2.01 operator names
)
\inf@bad=\count88
LaTeX Info: Redefining \frac on input line 211.
\uproot@=\count89
\leftroot@=\count90
LaTeX Info: Redefining \overline on input line 307.
\classnum@=\count91
\DOTSCASE@=\count92
LaTeX Info: Redefining \ldots on input line 379.
LaTeX Info: Redefining \dots on input line 382.
LaTeX Info: Redefining \cdots on input line 467.
\Mathstrutbox@=\box26
\strutbox@=\box27
\big@size=\dimen105
LaTeX Font Info:    Redeclaring font encoding OML on input line 567.
LaTeX Font Info:    Redeclaring font encoding OMS on input line 568.
\macc@depth=\count93
\c@MaxMatrixCols=\count94
\dotsspace@=\muskip10
\c@parentequation=\count95
\dspbrk@lvl=\count96
\tag@help=\toks17
\row@=\count97
\column@=\count98
\maxfields@=\count99
\andhelp@=\toks18
\eqnshift@=\dimen106
\alignsep@=\dimen107
\tagshift@=\dimen108
\tagwidth@=\dimen109
\totwidth@=\dimen110
\lineht@=\dimen111
\@envbody=\toks19
\multlinegap=\skip44
\multlinetaggap=\skip45
\mathdisplay@stack=\toks20
LaTeX Info: Redefining \[ on input line 2666.
LaTeX Info: Redefining \] on input line 2667.
) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
Package: amssymb 2002/01/22 v2.2d
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty
Package: amsfonts 2001/10/25 v2.2f
\symAMSa=\mathgroup4
\symAMSb=\mathgroup5
LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
(Font)                  U/euf/m/n --> U/euf/b/n on input line 132.
)) (/usr/share/texmf-texlive/tex/latex/base/alltt.sty
Package: alltt 1997/06/16 v2.0g defines alltt environment
) (/usr/share/texmf-texlive/tex/latex/tools/array.sty
Package: array 2005/08/23 v2.4b Tabular extension package (FMi)
\col@sep=\dimen112
\extrarowheight=\dimen113
\NC@list=\toks21
\extratabsurround=\skip46
\backup@length=\skip47
) (/usr/share/texmf-texlive/tex/latex/tools/indentfirst.sty
Package: indentfirst 1995/11/23 v1.03 Indent first paragraph (DPC)
) (/usr/share/texmf-texlive/tex/latex/footmisc/footmisc.sty
Package: footmisc 2005/03/17 v5.3d a miscellany of footnote facilities
\FN@temptoken=\toks22
\footnotemargin=\dimen114
\c@pp@next@reset=\count100
\c@@fnserial=\count101
Package footmisc Info: Declaring symbol style bringhurst on input line 817.
Package footmisc Info: Declaring symbol style chicago on input line 818.
Package footmisc Info: Declaring symbol style wiley on input line 819.
Package footmisc Info: Declaring symbol style lamport-robust on input line 823.

Package footmisc Info: Declaring symbol style lamport* on input line 831.
Package footmisc Info: Declaring symbol style lamport*-robust on input line 840
.
) (/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty
Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty
Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
\KV@toks@=\toks23
) (/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty
Package: graphics 2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty
Package: trig 1999/03/16 v1.09 sin cos tan (DPC)
) (/etc/texmf/tex/latex/config/graphics.cfg
File: graphics.cfg 2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
)
Package graphics Info: Driver file: pdftex.def on input line 90.
(/usr/share/texmf-texlive/tex/latex/pdftex-def/pdftex.def
File: pdftex.def 2007/01/08 v0.04d Graphics/color for pdfTeX
\Gread@gobject=\count102
))
\Gin@req@height=\dimen115
\Gin@req@width=\dimen116
) (/usr/share/texmf-texlive/tex/latex/tools/calc.sty
Package: calc 2005/08/06 v4.2 Infix arithmetic (KKT,FJ)
\calc@Acount=\count103
\calc@Bcount=\count104
\calc@Adimen=\dimen117
\calc@Bdimen=\dimen118
\calc@Askip=\skip48
\calc@Bskip=\skip49
LaTeX Info: Redefining \setlength on input line 75.
LaTeX Info: Redefining \addtolength on input line 76.
\calc@Ccount=\count105
\calc@Cskip=\skip50
) (/usr/share/texmf-texlive/tex/latex/fancyhdr/fancyhdr.sty
\fancy@headwidth=\skip51
\f@ncyO@elh=\skip52
\f@ncyO@erh=\skip53
\f@ncyO@olh=\skip54
\f@ncyO@orh=\skip55
\f@ncyO@elf=\skip56
\f@ncyO@erf=\skip57
\f@ncyO@olf=\skip58
\f@ncyO@orf=\skip59
) (/usr/share/texmf-texlive/tex/latex/geometry/geometry.sty
Package: geometry 2002/07/08 v3.2 Page Geometry
\Gm@cnth=\count106
\Gm@cntv=\count107
\c@Gm@tempcnt=\count108
\Gm@bindingoffset=\dimen119
\Gm@wd@mp=\dimen120
\Gm@odd@mp=\dimen121
\Gm@even@mp=\dimen122
\Gm@dimlist=\toks24
(/usr/share/texmf-texlive/tex/xelatex/xetexconfig/geometry.cfg)) (/usr/share/te
xmf-texlive/tex/latex/hyperref/hyperref.sty
Package: hyperref 2007/02/07 v6.75r Hypertext links for LaTeX
\@linkdim=\dimen123
\Hy@linkcounter=\count109
\Hy@pagecounter=\count110
(/usr/share/texmf-texlive/tex/latex/hyperref/pd1enc.def
File: pd1enc.def 2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
) (/etc/texmf/tex/latex/config/hyperref.cfg
File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
) (/usr/share/texmf-texlive/tex/latex/oberdiek/kvoptions.sty
Package: kvoptions 2006/08/22 v2.4 Connects package keyval with LaTeX options (
HO)
)
Package hyperref Info: Option `hyperfootnotes' set `false' on input line 2238.
Package hyperref Info: Option `bookmarks' set `true' on input line 2238.
Package hyperref Info: Option `linktocpage' set `false' on input line 2238.
Package hyperref Info: Option `pdfdisplaydoctitle' set `true' on input line 223
8.
Package hyperref Info: Option `pdfpagelabels' set `true' on input line 2238.
Package hyperref Info: Option `bookmarksopen' set `true' on input line 2238.
Package hyperref Info: Option `colorlinks' set `true' on input line 2238.
Package hyperref Info: Hyper figures OFF on input line 2288.
Package hyperref Info: Link nesting OFF on input line 2293.
Package hyperref Info: Hyper index ON on input line 2296.
Package hyperref Info: Plain pages OFF on input line 2303.
Package hyperref Info: Backreferencing OFF on input line 2308.
Implicit mode ON; LaTeX internals redefined
Package hyperref Info: Bookmarks ON on input line 2444.
(/usr/share/texmf-texlive/tex/latex/ltxmisc/url.sty
\Urlmuskip=\muskip11
Package: url 2005/06/27  ver 3.2  Verb mode for urls, etc.
)
LaTeX Info: Redefining \url on input line 2599.
\Fld@menulength=\count111
\Field@Width=\dimen124
\Fld@charsize=\dimen125
\Choice@toks=\toks25
\Field@toks=\toks26
Package hyperref Info: Hyper figures OFF on input line 3102.
Package hyperref Info: Link nesting OFF on input line 3107.
Package hyperref Info: Hyper index ON on input line 3110.
Package hyperref Info: backreferencing OFF on input line 3117.
Package hyperref Info: Link coloring ON on input line 3120.
\Hy@abspage=\count112
\c@Item=\count113
)
*hyperref using driver hpdftex*
(/usr/share/texmf-texlive/tex/latex/hyperref/hpdftex.def
File: hpdftex.def 2007/02/07 v6.75r Hyperref driver for pdfTeX
\Fld@listcount=\count114
)
\TmpLen=\skip60
(./36856-t.aux)
\openout1 = `36856-t.aux'.

LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 567.
LaTeX Font Info:    ... okay on input line 567.
LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 567.
LaTeX Font Info:    ... okay on input line 567.
LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 567.
LaTeX Font Info:    ... okay on input line 567.
LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 567.
LaTeX Font Info:    ... okay on input line 567.
LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 567.
LaTeX Font Info:    ... okay on input line 567.
LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 567.
LaTeX Font Info:    ... okay on input line 567.
LaTeX Font Info:    Checking defaults for PD1/pdf/m/n on input line 567.
LaTeX Font Info:    ... okay on input line 567.
(/usr/share/texmf/tex/context/base/supp-pdf.tex
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count115
\scratchdimen=\dimen126
\scratchbox=\box28
\nofMPsegments=\count116
\nofMParguments=\count117
\everyMPshowfont=\toks27
\MPscratchCnt=\count118
\MPscratchDim=\dimen127
\MPnumerator=\count119
\everyMPtoPDFconversion=\toks28
)
-------------------- Geometry parameters
paper: class default
landscape: --
twocolumn: --
twoside: true
asymmetric: --
h-parts: 79.49698pt, 415.55249pt, 119.24553pt
v-parts: 57.81598pt, 650.43pt, 86.72401pt
hmarginratio: 2:3
vmarginratio: 2:3
lines: --
heightrounded: --
bindingoffset: 0.0pt
truedimen: --
includehead: --
includefoot: --
includemp: --
driver: pdftex
-------------------- Page layout dimensions and switches
\paperwidth  614.295pt
\paperheight 794.96999pt
\textwidth  415.55249pt
\textheight 650.43pt
\oddsidemargin  7.22699pt
\evensidemargin 46.97554pt
\topmargin  -46.3278pt
\headheight 12.0pt
\headsep    19.8738pt
\footskip   30.0pt
\marginparwidth 98.0pt
\marginparsep   7.0pt
\columnsep  10.0pt
\skip\footins  10.8pt plus 4.0pt minus 2.0pt
\hoffset 0.0pt
\voffset 0.0pt
\mag 1000
\@twosidetrue \@mparswitchtrue 
(1in=72.27pt, 1cm=28.45pt)
-----------------------
(/usr/share/texmf-texlive/tex/latex/graphics/color.sty
Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC)
(/etc/texmf/tex/latex/config/color.cfg
File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive
)
Package color Info: Driver file: pdftex.def on input line 130.
)
Package hyperref Info: Link coloring ON on input line 567.
(/usr/share/texmf-texlive/tex/latex/hyperref/nameref.sty
Package: nameref 2006/12/27 v2.28 Cross-referencing by name of section
(/usr/share/texmf-texlive/tex/latex/oberdiek/refcount.sty
Package: refcount 2006/02/20 v3.0 Data extraction from references (HO)
)
\c@section@level=\count120
)
LaTeX Info: Redefining \ref on input line 567.
LaTeX Info: Redefining \pageref on input line 567.
(./36856-t.out) (./36856-t.out)
\@outlinefile=\write3
\openout3 = `36856-t.out'.

LaTeX Font Info:    Try loading font information for T1+cmtt on input line 573.

(/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd
File: t1cmtt.fd 1999/05/25 v2.5h Standard LaTeX font definitions
)
LaTeX Font Info:    Try loading font information for U+msa on input line 598.
(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2002/01/19 v2.2g AMS font definitions
)
LaTeX Font Info:    Try loading font information for U+msb on input line 598.
(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2002/01/19 v2.2g AMS font definitions
) [1

{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [1


] [2] [3


] [4] [5] [6


] [7] [8


] [1



] [2

] [3

] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Overfull \hbox (10.88365pt too wide) detected at line 1729
 []$[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 G [] \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/
n/12 2(\OML/cmm/m/it/12 EG \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 F[]\OT1/cmr/m/
n/12 ) [] \OML/cmm/m/it/12 :  $ 
 []

[19] [20]
Overfull \hbox (6.93011pt too wide) detected at line 1845
 []$[] \OMS/cmsy/m/n/12 ^^@ [][]\OML/cmm/m/it/12 ^^Nx \OMS/cmsy/m/n/12 ^^A \OML
/cmm/m/it/12 d[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 ^^Ny \OMS/cmsy/m/n/12 ^^A \
OML/cmm/m/it/12 d[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 ^^Nz \OMS/cmsy/m/n/12 ^^
A \OML/cmm/m/it/12 d[][]:[]  $ 
 []

[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
Underfull \vbox (badness 7099) has occurred while \output is active []

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43


] [44] [45] [46] [47] <./images/051.pdf, id=572, 593.21625pt x 183.68625pt>
File: ./images/051.pdf Graphic file (type pdf)
<use ./images/051.pdf> [48

 <./images/051.pdf>]
Underfull \vbox (badness 1033) has occurred while \output is active []

[49] [50] <./images/055.pdf, id=652, 397.485pt x 271.0125pt>
File: ./images/055.pdf Graphic file (type pdf)
<use ./images/055.pdf> [51] [52 <./images/055.pdf>] [53] [54]
Underfull \vbox (badness 4467) has occurred while \output is active []

[55] [56]
Overfull \hbox (11.82814pt too wide) detected at line 3858
 []$[][] []  $ 
 []

[57] [58] [59] [60]
Underfull \vbox (badness 10000) has occurred while \output is active []

[61] [62] [63] [64] [65]
Underfull \vbox (badness 5893) has occurred while \output is active []

[66]
Overfull \hbox (6.7065pt too wide) in paragraph at lines 4668--4668
[] 
 []

[67]
Overfull \hbox (13.14645pt too wide) detected at line 4751
 []$[][] []  $ 
 []


Overfull \hbox (7.81683pt too wide) detected at line 4766
 []$[][] []  $ 
 []

[68] [69] [70]
Overfull \hbox (5.58578pt too wide) detected at line 4938
 []$[][] []  $ 
 []

[71] [72] [73] [74] [75] [76


] [77] [78

] [79

] [80] [81] [82] [83] [84] [85] [86] [87]
LaTeX Font Info:    Try loading font information for U+euf on input line 5824.
(/usr/share/texmf-texlive/tex/latex/amsfonts/ueuf.fd
File: ueuf.fd 2002/01/19 v2.2g AMS font definitions
) [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] <./images/102.pdf, id=
1096, 403.5075pt x 140.525pt>
File: ./images/102.pdf Graphic file (type pdf)
<use ./images/102.pdf> [99] <./images/103.pdf, id=1107, 169.63374pt x 128.48pt>
File: ./images/103.pdf Graphic file (type pdf)
<use ./images/103.pdf> [100 <./images/102.pdf> <./images/103.pdf>] [101] [102] 
[103] [104] [105] [106]
Overfull \hbox (1.40654pt too wide) in paragraph at lines 6952--6952
[] 
 []


Overfull \hbox (2.0021pt too wide) detected at line 6970
[] \OML/cmm/m/it/10.95 ds d^^R:
 []

[107] [108

] [109] <./images/114.pdf, id=1252, 163.61125pt x 126.4725pt>
File: ./images/114.pdf Graphic file (type pdf)
<use ./images/114.pdf>
File: ./images/114.pdf Graphic file (type pdf)
<use ./images/114.pdf> [110 <./images/114.pdf>] [111] [112


] [113



] [114


] [115] [116] [117] [118] [119]
Overfull \hbox (1.53864pt too wide) in paragraph at lines 7571--7573
[]\T1/cmr/m/it/12 Leçons sur la théorie générale des sur-faces et les ap-pli-ca
-tions géométriques
 []

[120] [121] [122] [123] [124] [125] [126] [127] [128] [129]
Overfull \hbox (1.5184pt too wide) in paragraph at lines 8198--8200
[]\T1/cmr/m/it/12 Leçons sur les ap-pli-ca-tions géométriques de l'analyse. \T1
/cmr/m/n/12 Paris, \T1/cmr/m/sc/12 vi \T1/cmr/m/n/12 + 251 pp.,
 []

[130] [131] [132] [133] [134]
Overfull \hbox (6.14713pt too wide) in paragraph at lines 8484--8486
[]\T1/cmr/m/it/12 Ueber die The-o-rie der auf einan-der Ab-wick-el-baren Ober-f
lächen. \T1/cmr/m/n/12 Festschrift
 []

[135] [136]
Underfull \vbox (badness 10000) has occurred while \output is active []

[1


]
Underfull \vbox (badness 10000) has occurred while \output is active []

[2]
Underfull \vbox (badness 10000) has occurred while \output is active []

[3]
Underfull \vbox (badness 10000) has occurred while \output is active []

[4]
Underfull \vbox (badness 10000) has occurred while \output is active []

[5]
Underfull \vbox (badness 10000) has occurred while \output is active []

[6] [7] (./36856-t.aux)

 *File List*
    book.cls    2005/09/16 v1.4f Standard LaTeX document class
   leqno.clo    1998/08/17 v1.1c Standard LaTeX option (left equation numbers)
    bk12.clo    2005/09/16 v1.4f Standard LaTeX file (size option)
inputenc.sty    2006/05/05 v1.1b Input encoding file
  latin1.def    2006/05/05 v1.1b Input encoding file
 fontenc.sty
   t1enc.def    2005/09/27 v1.99g Standard LaTeX file
  ifthen.sty    2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
 amsmath.sty    2000/07/18 v2.13 AMS math features
 amstext.sty    2000/06/29 v2.01
  amsgen.sty    1999/11/30 v2.0
  amsbsy.sty    1999/11/29 v1.2d
  amsopn.sty    1999/12/14 v2.01 operator names
 amssymb.sty    2002/01/22 v2.2d
amsfonts.sty    2001/10/25 v2.2f
   alltt.sty    1997/06/16 v2.0g defines alltt environment
   array.sty    2005/08/23 v2.4b Tabular extension package (FMi)
indentfirst.sty    1995/11/23 v1.03 Indent first paragraph (DPC)
footmisc.sty    2005/03/17 v5.3d a miscellany of footnote facilities
graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
  keyval.sty    1999/03/16 v1.13 key=value parser (DPC)
graphics.sty    2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
    trig.sty    1999/03/16 v1.09 sin cos tan (DPC)
graphics.cfg    2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
  pdftex.def    2007/01/08 v0.04d Graphics/color for pdfTeX
    calc.sty    2005/08/06 v4.2 Infix arithmetic (KKT,FJ)
fancyhdr.sty    
geometry.sty    2002/07/08 v3.2 Page Geometry
geometry.cfg
hyperref.sty    2007/02/07 v6.75r Hypertext links for LaTeX
  pd1enc.def    2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
kvoptions.sty    2006/08/22 v2.4 Connects package keyval with LaTeX options (HO
)
     url.sty    2005/06/27  ver 3.2  Verb mode for urls, etc.
 hpdftex.def    2007/02/07 v6.75r Hyperref driver for pdfTeX
supp-pdf.tex
   color.sty    2005/11/14 v1.0j Standard LaTeX Color (DPC)
   color.cfg    2007/01/18 v1.5 color configuration of teTeX/TeXLive
 nameref.sty    2006/12/27 v2.28 Cross-referencing by name of section
refcount.sty    2006/02/20 v3.0 Data extraction from references (HO)
 36856-t.out
 36856-t.out
  t1cmtt.fd    1999/05/25 v2.5h Standard LaTeX font definitions
    umsa.fd    2002/01/19 v2.2g AMS font definitions
    umsb.fd    2002/01/19 v2.2g AMS font definitions
./images/051.pdf
./images/055.pdf
    ueuf.fd    2002/01/19 v2.2g AMS font definitions
./images/102.pdf
./images/103.pdf
./images/114.pdf
./images/114.pdf
 ***********

 ) 
Here is how much of TeX's memory you used:
 5707 strings out of 94074
 72349 string characters out of 1165154
 148739 words of memory out of 1500000
 8292 multiletter control sequences out of 10000+50000
 31015 words of font info for 79 fonts, out of 1200000 for 2000
 648 hyphenation exceptions out of 8191
 29i,21n,43p,258b,491s stack positions out of 5000i,500n,6000p,200000b,5000s
{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc}</usr/share/texmf-
texlive/fonts/type1/bluesky/cm/cmex10.pfb></usr/share/texmf-texlive/fonts/type1
/bluesky/cm/cmmi10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi12.
pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi8.pfb></usr/share/texm
f-texlive/fonts/type1/bluesky/cm/cmr10.pfb></usr/share/texmf-texlive/fonts/type
1/bluesky/cm/cmr12.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmr7.pf
b></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmr8.pfb></usr/share/texmf-t
exlive/fonts/type1/bluesky/cm/cmsy10.pfb></usr/share/texmf-texlive/fonts/type1/
bluesky/cm/cmsy6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmsy7.pfb
></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmsy8.pfb></usr/share/texmf-t
exlive/fonts/type1/bluesky/ams/eufm10.pfb></usr/share/texmf/fonts/type1/public/
cm-super/sfbx1200.pfb></usr/share/texmf/fonts/type1/public/cm-super/sfcc0800.pf
b></usr/share/texmf/fonts/type1/public/cm-super/sfcc1000.pfb></usr/share/texmf/
fonts/type1/public/cm-super/sfcc1200.pfb></usr/share/texmf/fonts/type1/public/c
m-super/sfcc1440.pfb></usr/share/texmf/fonts/type1/public/cm-super/sfcc2074.pfb
></usr/share/texmf/fonts/type1/public/cm-super/sfrm0800.pfb></usr/share/texmf/f
onts/type1/public/cm-super/sfrm1000.pfb></usr/share/texmf/fonts/type1/public/cm
-super/sfrm1095.pfb></usr/share/texmf/fonts/type1/public/cm-super/sfrm1200.pfb>
</usr/share/texmf/fonts/type1/public/cm-super/sfrm1440.pfb></usr/share/texmf/fo
nts/type1/public/cm-super/sfrm2074.pfb></usr/share/texmf/fonts/type1/public/cm-
super/sfti0800.pfb></usr/share/texmf/fonts/type1/public/cm-super/sfti1000.pfb><
/usr/share/texmf/fonts/type1/public/cm-super/sfti1095.pfb></usr/share/texmf/fon
ts/type1/public/cm-super/sfti1200.pfb></usr/share/texmf/fonts/type1/public/cm-s
uper/sftt0900.pfb></usr/share/texmf/fonts/type1/public/cm-super/sftt1095.pfb>
Output written on 36856-t.pdf (153 pages, 1093886 bytes).
PDF statistics:
 1653 PDF objects out of 1728 (max. 8388607)
 589 named destinations out of 1000 (max. 131072)
 154 words of extra memory for PDF output out of 10000 (max. 10000000)

