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 $z$  may be regarded as a function of $x,$  $y,$ 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 $t,$  $u$

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

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 $dt,$  $du$ 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,

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