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 $C$ be another point on the same shortest line $DB,$  for which point $p$  remains the same as for the point $B,$  and $q',$  $r',$  $\phi',$  $\psi',$  $S'$  have the same meanings as $q,$  $r,$  $\phi,$  $\psi,$  $S$  have for the point $B.$  There will thus be a triangle between the points $A,$  $B,$  $C,$  whose angles we denote by $A,$  $B,$  $C,$  the sides opposite these angles by $a,$  $b,$  $c,$  and the area by $\sigma.$  We represent the measure of curvature at the points $A,$  $B,$  $C$  by $\alpha,$  $\beta,$  $\gamma$  respectively. And then supposing (which is permissible) that the quantities $p,$  $q,$  $q - q'$ 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 $\sigma$ by a series. By changing in [7] each of the quantities that refer to $B$ into those that refer to $C,$  we obtain a formula for $S'.$  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 $m,$  $p,$  $q$ were what $n,$  $q,$  $p$  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 $p,$  $q$ refer to the point $B,$

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