Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/103

$${\begin{alignedat}{6}&\cos L&&(1)=x,&&\cos L&&(2)=y,&&\cos L&&(3)=z,\\&\cos L'&&(1)=x',&&\cos L'&&(2)=y',&&\cos L'&&(3)=z',\\&\cos L''&&(1)=x'',\quad &&\cos L''&&(2)=y'',\quad &&\cos L''&&(3)=z''.\end{alignedat}}$$

We assume that the points are so arranged that they run around the triangle included by them in the same sense as the points ${\textstyle (1),}$ ${\textstyle (2),}$ ${\textstyle (3).}$ Further, let ${\textstyle \lambda }$ be that pole of the great circle ${\textstyle L'L''}$ which lies on the same side as ${\textstyle L.}$ We then have, from the above lemma,

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

Therefore, if we multiply these equations by ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ respectively, and add the products, we obtain

$$xy'z''+x'y''z+x''yz'-xy''z'-x'yz''-x''y'z=\sin L'L''.\cos \lambda L,$$

wherefore, we can write also, according to well known principles of spherical trigonometry,

$${\begin{alignedat}{2}\sin L'L''.&\sin LL''&&.\sin L'\\=\sin L'L''.&\sin LL'&&.\sin L''\\=\sin L'L''.&\sin L'L''&&.\sin L,\end{alignedat}}$$

if ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ denote the three angles of the spherical triangle. At the same time we easily see that this value is one-sixth of the pyramid whose angular points are the centre of the sphere and the three points ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ (and indeed positive, if etc.).

8.

The nature of a curved surface is defined by an equation between the coordinates of its points, which we represent by

$$f(x,y,z)=0.$$

Let the total differential of ${\textstyle f(x,y,z)}$ be

$$P\,dx+Q\,dy+R\,dz,$$

where ${\textstyle P,}$ ${\textstyle Q,}$ ${\textstyle R}$ are functions of ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z.}$ We shall always distinguish two sides of the surface, one of which we shall call the upper, and the other the lower. Generally speaking, on passing through the surface the value of ${\textstyle f}$ changes its sign, so that, as long as the continuity is not interrupted, the values are positive on one side and negative on the other.