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 ${\textstyle p}$ denotes in a general manner the length of the shortest line drawn from a fixed point ${\textstyle A}$ to any other point whatever of the surface, and ${\textstyle q}$ the angle that the first element of this line makes with the first element of another given shortest line going out from ${\textstyle A.}$ Let ${\textstyle B}$ be a definite point in the latter line, for which ${\textstyle q=0,}$ and ${\textstyle C}$ another definite point of the surface, at which we denote the value of ${\textstyle q}$ simply by ${\textstyle A.}$ Let us suppose the points ${\textstyle B,}$ ${\textstyle C}$ joined by a shortest line, the parts of which, measured from ${\textstyle B,}$ we denote in a general way, as in Art. 18, by ${\textstyle s;}$ and, as in the same article, let us denote by ${\textstyle \theta }$ the angle which any element ${\textstyle ds}$ makes with the element ${\textstyle dp;}$ finally, let us denote by ${\textstyle \theta ^{\circ },}$ ${\textstyle \theta '}$ the values of the angle ${\textstyle \theta }$ at the points ${\textstyle B,}$ ${\textstyle C.}$ We have thus on the curved surface a triangle formed by shortest lines. The angles of this triangle at ${\textstyle B}$ and ${\textstyle C}$ we shall denote simply by the same letters, and ${\textstyle B}$ will be equal to ${\textstyle 180^{\circ }-\theta ,}$ ${\textstyle C}$ to ${\textstyle \theta '}$ 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 ${\textstyle 57^{\circ }\,17'\,45'',}$ 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 ${\textstyle 2\pi }$ denotes the circumference of the sphere. Let us now examine the integral curvature of this triangle, which is equal to

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

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

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

over the whole surface of the triangle. Let us begin by integration with respect to ${\textstyle p,}$ 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 ${\textstyle q,}$ ${\textstyle q+dq}$ of the second indeterminate. Since this inte-