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 ${\textstyle p,}$ ${\textstyle q}$ their most general meaning, we take for ${\textstyle p',}$ ${\textstyle q'}$ the quantities denoted in Art. 15 by ${\textstyle r,}$ ${\textstyle \phi .}$ We shall use ${\textstyle r,}$ ${\textstyle \phi }$ here also in such a way that, for any point whatever on the surface, ${\textstyle r}$ will be the shortest distance from a fixed point, and ${\textstyle \phi }$ the angle at this point between the first element of ${\textstyle r}$ 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 ${\textstyle \alpha ,}$ ${\textstyle \beta ,}$ ${\textstyle \gamma ,}$ ${\textstyle \delta }$ give

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