Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/19

But as there are for each great circle two branches going out from the point ${\textstyle A,}$ these two branches form at this point two angles whose sum is ${\textstyle 180^{\circ }.}$ But our analysis shows that those branches are to be taken whose directions are in the sense from the point ${\textstyle L}$ to ${\textstyle L',}$ and from the point ${\textstyle L''}$ to ${\textstyle L''';}$ and since great circles intersect in two points, it is clear that either of the two points can be chosen arbitrarily. Also, instead of the angle ${\textstyle A,}$ we can take the arc between the poles of the great circles of which the arcs ${\textstyle LL',}$ ${\textstyle L''L'''}$ are parts. But it is evident that those poles are to be chosen which are similarly placed with respect to these arcs; that is to say, when we go from ${\textstyle L}$ to ${\textstyle L'}$ and from ${\textstyle L''}$ to ${\textstyle L''',}$ both of the two poles are to be on the right, or both on the left.

VII. Let ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ be the three points on the sphere and set, for brevity,

$${\begin{alignedat}{3}&\cos(1)L&&=x,\quad &&\cos(2)L&&=y,\quad &&\cos(3)L&&=z\\&\cos(1)L'&&=x',&&\cos(2)L'&&=y',&&\cos(3)L'&&=z'\\&\cos(1)L''&&=x'',&&\cos(2)L''&&=y'',&&\cos(3)L''&&=z''\\\end{alignedat}}$$

and also

$$xy'z''+x'y''z+x''yz'-xy''z'-x'yz''-x''y'z=\Delta$$

Let ${\textstyle \lambda }$ denote the pole of the great circle of which ${\textstyle LL'}$ is a part, this pole being the one that is placed in the same position with respect to this arc as the point ${\textstyle (1)}$ is with respect to the arc ${\textstyle (2)(3).}$ Then we shall have, by the preceding theorem,

$$yz'-y'z=\cos(1)\lambda .\sin(2)(3).\sin LL',$$

or, because ${\textstyle (2)(3)=90^{\circ },}$

$${\begin{aligned}yz'-y'z&=\cos(1)\lambda .\sin LL',\end{aligned}}$$

and similarly,

$${\begin{aligned}zx'-z'x&=\cos(2)\lambda .\sin LL'\\xy'-x'y&=\cos(3)\lambda .\sin LL'\end{aligned}}$$

Multiplying these equations by ${\textstyle x'',}$ ${\textstyle y'',}$ ${\textstyle z''}$ respectively, and adding, we obtain, by means of the second of the theorems deduced in V,

$$\Delta =\cos \lambda L''.\sin LL'$$

Now there are three cases to be distinguished. First, when ${\textstyle L''}$ lies on the great circle of which the arc ${\textstyle LL'}$ is a part, we shall have ${\textstyle \lambda L''=90^{\circ },}$ and consequently, ${\textstyle \Delta =0.}$ If ${\textstyle L''}$ does not lie on that great circle, the second case will be when ${\textstyle L''}$ is on the same side as ${\textstyle \lambda ;}$ the third case when they are on opposite sides. In the last two cases the points ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ will form a spherical triangle, and in the second case these points will lie in the same order as the points ${\textstyle (1),}$ ${\textstyle (2),}$ ${\textstyle (3),}$ and in the opposite order in the third case.