Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/54

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$$\begin{aligned} \gamma &= -2f^{\circ} - 2f'p - 6g^{\circ}q' - 2f''p^{2} - 6g'pq' - (12h^{\circ} - 2{f^{\circ}}^{2})q'^{2} - \text{etc.} \\ \alpha &= -2f^{\circ} \end{aligned}$$

Introducing these measures of curvature into the expression for $\sigma,$ we obtain the following expression, exact to quantities of the sixth order (exclusive):

$$\begin{aligned} \quad\; \sigma = \tfrac{1}{2}ac \sin B \bigl(1 &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} + 3qq' + 3q'^{2}) \\          &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} + 6qq' + 3q'^{2}) \\          &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} + \phantom{0} qq' + 4q'^{2})\bigr) \end{aligned}$$

The same precision will remain, if for $p,$  $q,$  $q'$ we substitute $c\sin B,$  $c\cos B,$  $c\cos B - a.$  This gives

[8] $$\begin{aligned} \sigma = \tfrac{1}{2}ac\sin B \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \phantom{0}9ac\cos B) \\          &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\          &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \phantom{0}9ac\cos B)\bigr) \end{aligned}$$

Since all expressions which refer to the line $AD$ drawn normal to $BC$  have disappeared from this equation, we may permute among themselves the points $A,$  $B,$  $C$  and the expressions that refer to them. Therefore we shall have, with the same precision,

[9] $$\begin{aligned} \sigma = \tfrac{1}{2}bc\sin A \bigl(1 &+ \tfrac{1}{120}\alpha(3b^{2} + 3c^{2} -  12bc\cos A) \\          &+ \tfrac{1}{120}\beta (3b^{2} + 4c^{2} - \phantom{0}9bc\cos A) \\          &+ \tfrac{1}{120}\gamma(4b^{2} + 3c^{2} - \phantom{0}9bc\cos A)\bigr) \\ \end{aligned}$$

[10] $$\begin{aligned} \sigma = \tfrac{1}{2}ab\sin C \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4b^{2} - \phantom{0}9ab\cos C) \\          &+ \tfrac{1}{120}\beta (4a^{2} + 3b^{2} - \phantom{0}9ab\cos C) \\          &+ \tfrac{1}{120}\gamma(3a^{2} + 3b^{2} -  12ab\cos C)\bigr) \end{aligned}$$

26.

The consideration of the rectilinear triangle whose sides are equal to $a,$  $b,$  $c$ is of great advantage. The angles of this triangle, which we shall denote by $A^{*},$  $B^{*},$  $C^{*},$ differ from the angles of the triangle on the curved surface, namely, from $A,$  $B,$  $C,$  by quantities of the second order; and it will be worth while to develop these differences accurately. However, it will be sufficient to show the first steps in these more tedious than difficult calculations.

Replacing in formulæ [1], [4], [5] the quantities that refer to $B$ by those that refer to $C,$  we get formulæ for $r'^{2},$  $r'\cos\phi',$  $r'\sin\phi'.$  Then the development of the expression