Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/43



20.

We pause to investigate the case in which we suppose that $p$  denotes in a general manner the length of the shortest line drawn from a fixed point $A$ to any other point whatever of the surface, and $q$  the angle that the first element of this line makes with the first element of another given shortest line going out from $A.$  Let $B$  be a definite point in the latter line, for which $q = 0,$  and $C$  another definite point of the surface, at which we denote the value of $q$  simply by $A.$  Let us suppose the points $B,$  $C$  joined by a shortest line, the parts of which, measured from $B,$  we denote in a general way, as in Art. 18, by $s;$ and, as in the same article, let us denote by $\theta$  the angle which any element $ds$  makes with the element $dp;$  finally, let us denote by $\theta^{\circ},$  $\theta'$  the values of the angle $\theta$  at the points $B,$  $C.$  We have thus on the curved surface a triangle formed by shortest lines. The angles of this triangle at $B$  and $C$ we shall denote simply by the same letters, and $B$  will be equal to $180^{\circ} - \theta,$  $C$  to $\theta'$  itself. But, since it is easily seen from our analysis that all the angles are supposed to be expressed, not in degrees, but by numbers, in such a way that the angle $57^{\circ}\, 17'\, 45'',$ to which corresponds an arc equal to the radius, is taken for the unit, we must set

$$\theta^{\circ} = \pi - B,\quad \theta' = C$$

where $2\pi$  denotes the circumference of the sphere. Let us now examine the integral curvature of this triangle, which is equal to

$$\int k\, d\sigma,$$

$d\sigma$  denoting a surface element of the triangle. Wherefore, since this element is expressed by $m\, dp. dq,$ we must extend the integral

$$\iint km\, dp. dq$$

over the whole surface of the triangle. Let us begin by integration with respect to $p,$ which, because

$$k = -\frac{1}{m}. \frac{\partial^{2} m}{\partial p^{2}},$$

gives

$$dq. \left(\text{const.} - \frac{\partial m}{\partial p}\right),$$

for the integral curvature of the area lying between the lines of the first system, to which correspond the values $q,$  $q + dq$ of the second indeterminate. Since this inte-