Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/41

$${\begin{aligned}+\int \delta p\left({\frac {{\frac {\partial E}{\partial p}}.dp^{2}+2{\frac {\partial F}{\partial p}}.dp.dq+{\frac {\partial G}{\partial p}}.dq^{2}}{2\,ds}}-d.{\frac {E\,dp+F\,dq}{ds}}\right)\end{aligned}}$$

and we know that what is included under the integral sign must vanish independently of ${\textstyle \delta p.}$ Thus we have

$${\begin{aligned}{\frac {\partial E}{\partial p}}.dp^{2}&+2{\frac {\partial F}{\partial p}}.dp.dq+{\frac {\partial G}{\partial p}}.dq^{2}=2\,ds.d.{\frac {E\,dp+F\,dq}{ds}}\\&=2\,ds.d.{\sqrt {E}}.\cos \theta \\&={\frac {ds.dE.\cos \theta }{\sqrt {E}}}-2\,ds.d\theta .{\sqrt {E}}.\sin \theta \\&={\frac {(E\,dp+F\,dq)\,dE}{E}}-2{\sqrt {EG-F^{2}}}.dq.d\theta \\&=\left({\frac {E\,dp+F\,dq}{E}}\right).\left({\frac {\partial E}{\partial p}}.dp+{\frac {\partial E}{\partial q}}.dq\right)-2{\sqrt {EG-F^{2}}}.dq.d\theta \end{aligned}}$$

This gives the following conditional equation for a shortest line:

$${\begin{array}{c}{\sqrt {EG-F^{2}}}.d\theta ={\frac {1}{2}}.{\frac {F}{E}}.{\frac {\partial E}{\partial p}}.dp+{\frac {1}{2}}.{\frac {F}{E}}.{\frac {\partial E}{\partial q}}.dq+{\frac {1}{2}}.{\frac {\partial E}{\partial q}}.dp\\-{\frac {\partial F}{\partial p}}.dp-{\frac {1}{2}}.{\frac {\partial G}{\partial p}}.dq\end{array}}$$

which can also be written

$${\sqrt {EG-F^{2}}}.d\theta ={\frac {1}{2}}.{\frac {F}{E}}.dE+{\frac {1}{2}}.{\frac {\partial E}{\partial q}}.dp-{\frac {\partial F}{\partial p}}.dp-{\frac {1}{2}}.{\frac {\partial G}{\partial p}}.dq$$

From this equation, by means of the equation

$$\cot \theta ={\frac {E}{\sqrt {EG-F^{2}}}}\cdot {\frac {dp}{dq}}+{\frac {F}{\sqrt {EG-F^{2}}}}$$

it is also possible to eliminate the angle ${\textstyle \theta ,}$ and to derive a differential equation of the second order between ${\textstyle p}$ and ${\textstyle q,}$ which, however, would become more complicated and less useful for applications than the preceding.

19.

The general formulæ, which we have derived in Arts. 11, 18 for the measure of curvature and the variation in the direction of a shortest line, become much simpler if the quantities ${\textstyle p,}$ ${\textstyle q}$ are so chosen that the lines of the first system cut everywhere