Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/31

Thus, inasmuch as ${\textstyle z}$ may be regarded as a function of ${\textstyle x,}$ ${\textstyle y,}$ we have

$${\begin{aligned}{\frac {\partial z}{\partial x}}&=t=-{\frac {A}{C}}\\{\frac {\partial z}{\partial y}}&=u=-{\frac {B}{C}}\end{aligned}}$$

Then from the formulæ

$$dx=a\,dp+a'\,dq,\quad dy=b\,dp+b'\,dq,$$

we have

$${\begin{alignedat}{4}&C\,dp=&&b'\,&&dx-a'\,&&dy\\&C\,dq=-&&b\,&&dx+a\,&&dy\end{alignedat}}$$

Thence we obtain for the total differentials of ${\textstyle t,}$ ${\textstyle u}$

$${\begin{aligned}C^{3}\,dt&=\left(A\,{\frac {\partial C}{\partial p}}-C\,{\frac {\partial A}{\partial p}}\right)(b'\,dx-a'\,dy)+\left(C\,{\frac {\partial A}{\partial q}}-A\,{\frac {\partial C}{\partial q}}\right)(b\,dx-a\,dy)\\C^{3}\,du&=\left(B\,{\frac {\partial C}{\partial p}}-C\,{\frac {\partial B}{\partial p}}\right)(b'\,dx-a'\,dy)+\left(C\,{\frac {\partial B}{\partial q}}-B\,{\frac {\partial C}{\partial q}}\right)(b\,dx-a\,dy)\end{aligned}}$$

If now we substitute in these formulæ

$${\begin{alignedat}{4}{\frac {\partial A}{\partial p}}&=c'\beta &&+b\gamma '&&-c\beta '&&-b'\gamma \\{\frac {\partial A}{\partial q}}&=c'\beta '&&+b\gamma ''&&-c\beta ''&&-b'\gamma '\\{\frac {\partial B}{\partial p}}&=a'\gamma &&+c\alpha '&&-a\gamma '&&-c'\alpha \\{\frac {\partial B}{\partial q}}&=a'\gamma '&&+c\alpha ''&&-a\gamma ''&&-c'\alpha '\\{\frac {\partial C}{\partial p}}&=b'\alpha &&+a\beta '&&-b\alpha '&&-a'\beta \\{\frac {\partial C}{\partial q}}&=b'\alpha '&&+a\beta ''&&-b\alpha ''&&-a'\beta '\end{alignedat}}$$

and if we note that the values of the differentials ${\textstyle dt,}$ ${\textstyle du}$ thus obtained must be equal, independently of the differentials ${\textstyle dx,}$ ${\textstyle dy,}$ to the quantities ${\textstyle T\,dx+U\,dy,}$ ${\textstyle U\,dx+V\,dy}$ respectively, we shall find, after some sufficiently obvious transformations,

$${\begin{aligned}C^{3}T&=\alpha Ab'^{2}+\beta Bb'^{2}+\gamma Cb'^{2}\\&\quad -2\alpha 'Abb'-2\beta 'Bbb'-2\gamma 'Cbb'{\phantom {(ba'-a'b)(ba'-a'b}}\\&\quad +\alpha ''Ab^{2}+\beta ''Bb^{2}+\gamma ''Cb^{2}\end{aligned}}$$