Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/104

The direction of the normal to the surface toward that side which we regard as the upper side is represented upon the auxiliary sphere by the point ${\textstyle L.}$ Let

$$\cos L(1)=X,\qquad \cos L(2)=Y,\qquad \cos L(3)=Z.$$

Also let ${\textstyle ds}$ denote an infinitely small line upon the surface; and, as its direction is denoted by the point ${\textstyle \lambda }$ on the sphere, let

$$\cos \lambda (1)=\xi ,\qquad \cos \lambda (2)=\eta ,\qquad \cos \lambda (3)=\zeta .$$

We then have

$$dx=\xi \,ds,\qquad dy=\eta \,ds,\qquad dz=\zeta \,ds,$$

therefore

$$P\xi +Q\eta +R\zeta =0,$$

and, since ${\textstyle \lambda L}$ must be equal to ${\textstyle 90^{\circ },}$ we have also

$$X\xi +Y\eta +Z\zeta =0.$$

Since ${\textstyle P,}$ ${\textstyle Q,}$ ${\textstyle R,}$ ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z}$ depend only on the position of the surface on which we take the element, and since these equations hold for every direction of the element on the surface, it is easily seen that ${\textstyle P,}$ ${\textstyle Q,}$ ${\textstyle R}$ must be proportional to ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z.}$ Therefore

$$P=X\mu ,\qquad Q=Y\mu ,\qquad R=Z\mu .$$

Therefore, since

$${\begin{array}{c}X^{2}+Y^{2}+Z^{2}=1;\\\mu =PX+QY+RZ\end{array}}$$

and

$$\mu ^{2}=P^{2}+Q^{2}+R^{2},$$

or

$$\mu =\pm {\sqrt {P^{2}+Q^{2}+R^{2}}}.$$

If we go out from the surface, in the direction of the normal, a distance equal to the element ${\textstyle \delta \rho ,}$ then we shall have

$$\delta x=X\,\delta \rho ,\qquad \delta y=Y\,\delta \rho ,\qquad \delta z=Z\,\delta \rho$$

and

$$\delta f=P\,\delta x+Q\,\delta y+R\,\delta z=\mu \,\delta \rho .$$

We see, therefore, how the sign of ${\textstyle \mu }$ depends on the change of sign of the value of ${\textstyle f}$ in passing from the lower to the upper side.

9.

Let us cut the curved surface by a plane through the point to which our notation refers; then we obtain a plane curve of which ${\textstyle ds}$ is an element, in connection with which we shall retain the above notation. We shall regard as the upper side of the plane that one on which the normal to the curved surface lies. Upon this plane