Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/111

"The measure of curvature is always expressed by means of a fraction whose numerator is unity and whose denominator is the product of the maximum and minimum radii of curvature in the planes passing through the normal."

12.

We will now investigate the nature of shortest lines upon curved surfaces. The nature of a curved line in space is determined, in general, in such a way that the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ of each point are regarded as functions of a single variable, which we shall call ${\textstyle w.}$ The length of the curve, measured from an arbitrary origin to this point, is then equal to

$$\int {\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}.dw.$$

If we allow the curve to change its position by an infinitely small variation, the variation of the whole length will then be

$${\begin{aligned}&\qquad \qquad =\int {\frac {{\frac {dx}{dw}}.d\,\delta x+{\frac {dy}{dw}}.d\,\delta y+{\frac {dz}{dw}}.d\,\delta z}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}}={\frac {{\frac {dx}{dw}}.\delta x+{\frac {dy}{dw}}.\delta y+{\frac {dz}{dw}}.\delta z}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}}\\&-\int \left\{\delta x.d{\frac {\frac {dx}{dw}}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}}\right.+\delta y.d{\frac {\frac {dy}{dw}}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}}\\&\qquad \qquad +\left.\delta z.d{\frac {\frac {dz}{dw}}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}}\right\}.\end{aligned}}$$

The expression under the integral sign must vanish in the case of a minimum, as we know. Since the curved line lies upon a given curved surface whose equation is

$$P\,dx+Q\,dy+R\,dz=0,$$

the equation between the variations ${\textstyle \delta x,}$ ${\textstyle \delta y,}$ ${\textstyle \delta z}$

$$P\,\delta z+Q\,\delta y+R\,\delta z=0$$

must also hold. From this, by means of well known principles, we easily conclude that the differentials