Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/53

25.

From the formulæ of the preceding article, which refer to a right triangle formed by shortest lines, we proceed to the general case. Let ${\textstyle C}$ be another point on the same shortest line ${\textstyle DB,}$ for which point ${\textstyle p}$ remains the same as for the point ${\textstyle B,}$ and ${\textstyle q',}$ ${\textstyle r',}$ ${\textstyle \phi ',}$ ${\textstyle \psi ',}$ ${\textstyle S'}$ have the same meanings as ${\textstyle q,}$ ${\textstyle r,}$ ${\textstyle \phi ,}$ ${\textstyle \psi ,}$ ${\textstyle S}$ have for the point ${\textstyle B.}$ There will thus be a triangle between the points ${\textstyle A,}$ ${\textstyle B,}$ ${\textstyle C,}$ whose angles we denote by ${\textstyle A,}$ ${\textstyle B,}$ ${\textstyle C,}$ the sides opposite these angles by ${\textstyle a,}$ ${\textstyle b,}$ ${\textstyle c,}$ and the area by ${\textstyle \sigma .}$ We represent the measure of curvature at the points ${\textstyle A,}$ ${\textstyle B,}$ ${\textstyle C}$ by ${\textstyle \alpha ,}$ ${\textstyle \beta ,}$ ${\textstyle \gamma }$ respectively. And then supposing (which is permissible) that the quantities ${\textstyle p,}$ ${\textstyle q,}$ ${\textstyle q-q'}$ are positive, we shall have

$${\begin{aligned}A&=\phi -\phi ',&B&=\psi ,&C&=\pi -\psi ',&&\\a&=q-q',&b&=r',&c&=r,&\sigma &=S-S'.\end{aligned}}$$

We shall first express the area ${\textstyle \sigma }$ by a series. By changing in [7] each of the quantities that refer to ${\textstyle B}$ into those that refer to ${\textstyle C,}$ we obtain a formula for ${\textstyle S'.}$ Whence we have, exact to quantities of the sixth order,

$${\begin{aligned}\sigma ={\tfrac {1}{2}}p(q-q'){\bigl (}1&-{\tfrac {1}{6}}f^{\circ }(p^{2}+q^{2}+qq'+q'^{2})\\&-{\tfrac {1}{60}}f'p(6p^{2}+7q^{2}+7qq'+7q'^{2})\\&-{\tfrac {1}{20}}g^{\circ }(q+q')(3p^{2}+4q^{2}+4q'^{2}){\bigr )}\end{aligned}}$$

This formula, by aid of series [2], namely,

$$c\sin B=p(1-{\tfrac {1}{3}}f^{\circ }q^{2}-{\tfrac {1}{4}}f'pq^{2}-{\tfrac {1}{2}}g^{\circ }q^{3}-{\text{etc.}})$$

can be changed into the following:

$${\begin{aligned}\sigma ={\tfrac {1}{2}}ac\sin B{\bigl (}1&-{\tfrac {1}{6}}f^{\circ }(p^{2}-q^{2}+qq'+q'^{2})\\&-{\tfrac {1}{60}}f'p(6p^{2}-8q^{2}+7qq'+7q'^{2})\\&-{\tfrac {1}{20}}g^{\circ }(3p^{2}q+3p^{2}q'-6p^{3}+4q^{2}q'+4qq'^{2}+4q'^{3}){\bigr )}\end{aligned}}$$

The measure of curvature for any point whatever of the surface becomes (by Art. 19, where ${\textstyle m,}$ ${\textstyle p,}$ ${\textstyle q}$ were what ${\textstyle n,}$ ${\textstyle q,}$ ${\textstyle p}$ are here)

$${\begin{aligned}k&=-{\frac {1}{n}}.{\frac {\partial ^{2}n}{\partial q^{2}}}=-{\frac {2f+6gq+12hq^{2}+{\text{etc.}}}{1+fq^{2}+{\text{etc.}}}}\\&=-2f-6gq-(12h-2f^{2})q^{2}-{\text{etc.}}\end{aligned}}$$

Therefore we have, when ${\textstyle p,}$ ${\textstyle q}$ refer to the point ${\textstyle B,}$

$$\beta =-2f^{\circ }-2f'p-6g^{\circ }q-2f''p^{2}-6g'pq-(12h^{\circ }-2{f^{\circ }}^{2})q^{2}-{\text{etc.}}$$