Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/37

Moreover, it is easily seen that

$${\sqrt {d\xi ^{2}+d\eta ^{2}+d\zeta ^{2}}}$$

is equal to the small arc on the sphere which measures the angle between the directions of the tangents at the beginning and at the end of the element ${\textstyle dr,}$ and is thus equal to ${\textstyle {\frac {dr}{\rho }},}$ if ${\textstyle \rho }$ denotes the radius of curvature of the shortest line at this point. Thus we shall have

$$\rho \,d\xi =X\,dr,\quad \rho \,d\eta =Y\,dr,\quad \rho \,d\zeta =Z\,dr$$

15.

Suppose that an infinite number of shortest lines go out from a given point ${\textstyle A}$ on the curved surface, and suppose that we distinguish these lines from one another by the angle that the first element of each of them makes with the first element of one of them which we take for the first. Let ${\textstyle \phi }$ be that angle, or, more generally, a function of that angle, and ${\textstyle r}$ the length of such a shortest line from the point ${\textstyle A}$ to the point whose coordinates are ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z.}$ Since to definite values of the variables ${\textstyle r,}$ ${\textstyle \phi }$ there correspond definite points of the surface, the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ can be regarded as functions of ${\textstyle r,}$ ${\textstyle \phi .}$ We shall retain for the notation ${\textstyle \lambda ,}$ ${\textstyle L,}$ ${\textstyle \xi ,}$ ${\textstyle \eta ,}$ ${\textstyle \zeta ,}$ ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z}$ the same meaning as in the preceding article, this notation referring to any point whatever on any one of the shortest lines.

All the shortest lines that are of the same length ${\textstyle r}$ will end on another line whose length, measured from an arbitrary initial point, we shall denote by ${\textstyle v.}$ Thus ${\textstyle v}$ can be regarded as a function of the indeterminates ${\textstyle r,}$ ${\textstyle \phi ,}$ and if ${\textstyle \lambda '}$ denotes the point on the sphere corresponding to the direction of the element ${\textstyle dv,}$ and also ${\textstyle \xi ',}$ ${\textstyle \eta ',}$ ${\textstyle \zeta '}$ denote the coordinates of this point with reference to the centre of the sphere, we shall have

$${\frac {\partial x}{\partial \phi }}=\xi '\cdot {\frac {\partial v}{\partial \phi }},\quad {\frac {\partial y}{\partial \phi }}=\eta '\cdot {\frac {\partial v}{\partial \phi }},\quad {\frac {\partial z}{\partial \phi }}=\zeta '\cdot {\frac {\partial v}{\partial \phi }}$$

From these equations and from the equations

$${\frac {\partial x}{\partial r}}=\xi ,\quad {\frac {\partial y}{\partial r}}=\eta ,\quad {\frac {\partial z}{\partial r}}=\zeta$$

we have

$${\frac {\partial x}{\partial r}}\cdot {\frac {\partial x}{\partial \phi }}+{\frac {\partial y}{\partial r}}\cdot {\frac {\partial y}{\partial \phi }}+{\frac {\partial z}{\partial r}}\cdot {\frac {\partial z}{\partial \phi }}=(\xi \xi '+\eta \eta '+\zeta \zeta ')\cdot {\frac {\partial v}{\partial \phi }}=\cos \lambda \lambda '\cdot {\frac {\partial v}{\partial \phi }}$$