Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/50

or

$${\begin{alignedat}{2}n=1+f^{\circ }q^{2}&+f'pq^{2}&&+f''p^{2}q^{2}+{\text{etc.}}\\&+g^{\circ }q^{3}&&+g'pq^{3}+{\text{etc.}}\\&&&+h^{\circ }q^{4}+{\text{etc. etc.}}\end{alignedat}}$$

24.

The equations of Art. 22 give, in our case,

$${\begin{aligned}&n\sin \psi ={\frac {\partial r}{\partial p}},\quad \cos \psi ={\frac {\partial r}{\partial q}},\quad -n\cos \psi =m.{\frac {\partial \phi }{\partial r}},\quad \sin \psi =m.{\frac {\partial \phi }{\partial q}},\\&n^{2}=n^{2}\left({\frac {\partial r}{\partial q}}\right)^{2}+\left({\frac {\partial r}{\partial p}}\right)^{2},\quad n^{2}.{\frac {\partial r}{\partial q}}.{\frac {\partial \phi }{\partial q}}+{\frac {\partial r}{\partial p}}.{\frac {\partial \phi }{\partial p}}=0\end{aligned}}$$

By the aid of these equations, the fifth and sixth of which are contained in the others, series can be developed for ${\textstyle r,}$ ${\textstyle \phi ,}$ ${\textstyle \psi ,}$ ${\textstyle m,}$ 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 ${\textstyle p,}$ ${\textstyle q}$ we must have

$$r^{2}=p^{2}+q^{2},$$

the series for ${\textstyle r^{2}}$ will begin with the terms ${\textstyle p^{2}+q^{2}.}$ We obtain the terms of higher order by the method of undetermined coefficients,^[1] by means of the equation

$$\left({\frac {1}{n}}.{\frac {\partial (r^{2})}{\partial p}}\right)^{2}+\left({\frac {\partial (r^{2})}{\partial q}}\right)^{2}=4r^{2}$$

Thus we have

[1]

${\textstyle \displaystyle {\begin{alignedat}{3}r^{2}&=p^{2}+{\tfrac {2}{3}}f^{\circ }p^{2}q^{2}&&+{\tfrac {1}{2}}f'p^{3}q^{2}&&+({\tfrac {2}{5}}f''-{\tfrac {4}{45}}{f^{\circ }}^{2})p^{4}q^{2}\quad {\text{etc.}}\\&+q^{2}&&+{\tfrac {1}{2}}g^{\circ }p^{2}q^{3}&&+{\tfrac {2}{5}}g'p^{3}q^{3}\\&&&&&+({\tfrac {2}{5}}h^{\circ }-{\tfrac {7}{45}}{f^{\circ }}^{2})p^{2}q^{4}\end{alignedat}}}$

Then we have, from the formula

$$r\sin \psi ={\frac {1}{2n}}.{\frac {\partial (r^{2})}{\partial p}},$$

[2]

${\textstyle \displaystyle {\begin{alignedat}{2}r\sin \psi =p-{\tfrac {1}{3}}f^{\circ }pq^{2}&-{\tfrac {1}{4}}f'p^{2}q^{2}&&-({\tfrac {1}{5}}f''+{\tfrac {8}{45}}{f^{\circ }}^{2})p^{3}q^{2}\quad {\text{etc.}}\\&-{\tfrac {1}{2}}g^{\circ }pq^{3}&&-{\tfrac {2}{5}}g'p^{2}q^{3}\\&&&-({\tfrac {3}{5}}h^{\circ }-{\tfrac {8}{45}}{f^{\circ }}^{2})pq^{4}\end{alignedat}}}$

1. We have thought it useless to give the calculation here, which can be somewhat abridged by certain artifices.