Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/47

 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

$$EG - F^{2} = (E'G' - F'^{2})(\alpha\delta - \beta\gamma)^{2}$$

and since, 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

$$\begin{alignedat}{1} &E\delta^{2} - 2F\gamma\delta + G\gamma^{2} = \frac{EG - F^{2}}{E'G' - F'^{2}}. E' \\ -&E\beta\delta + F(\alpha\delta + \beta\gamma) - G\alpha\gamma = \frac{EG - F^{2}}{E'G' - F'^{2}}. F' \\ &E\beta^{2} - 2F\alpha\beta + G\alpha^{2} = \frac{EG - F^{2}}{E'G' - F'^{2}}. G' \end{alignedat}$$

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

$$E' = 1,\quad F' = 0,\quad \omega' = 90^{\circ}.$$

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

$$\begin{alignedat}{2} &\sqrt{E}. \cos(\omega - \psi) = \frac{\partial r}{\partial p} \quad.&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(1)} \\ &\sqrt{G}. \cos \psi = \frac{\partial r}{\partial q} \qquad.\quad.&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(2)} \end{alignedat}$$