Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/26

The double area of this triangle will be expressed by the formula

$$dx.\delta y-dy.\delta x$$

and this will be in a positive or negative form according as the position of the side from the first point to the third, with respect to the side from the first point to the second, is similar or opposite to the position of the ${\textstyle y}$-axis of coordinates with respect to the ${\textstyle x}$-axis of coordinates.

In like manner, if the coordinates of the three points which form the projection of the corresponding element on the sphere, from the centre of the sphere as origin, are

$${\begin{aligned}&X,&&Y\\&X+dX,\quad &&Y+dY\\&X+\delta X,\quad &&Y+\delta Y\end{aligned}}$$

the double area of this projection will be expressed by

$$dX.\delta Y-dY.\delta X$$

and the sign of this expression is determined in the same manner as above. Wherefore the measure of curvature at this point of the curved surface will be

$$k={\frac {dX.\delta Y-dY.\delta X}{dx.\delta y-dy.\delta x}}$$

If now we suppose the nature of the curved surface to be defined according to the third method considered in Art. 4, ${\textstyle X}$ and ${\textstyle Y}$ will be in the form of functions of the quantities ${\textstyle x,}$ ${\textstyle y.}$ We shall have, therefore,

$${\begin{aligned}dX&={\frac {\partial X}{\partial x}}\,dx&&+{\frac {\partial X}{\partial y}}\,dy\\\delta X&={\frac {\partial X}{\partial x}}\,\delta x&&+{\frac {\partial X}{\partial y}}\,\delta y\\dY&={\frac {\partial Y}{\partial x}}\,dx&&+{\frac {\partial Y}{\partial y}}\,dy\\\delta Y&={\frac {\partial Y}{\partial x}}\,\delta x&&+{\frac {\partial Y}{\partial y}}\,\delta y\end{aligned}}$$

When these values have been substituted, the above expression becomes

$$k={\frac {\partial X}{\partial x}}.{\frac {\partial Y}{\partial y}}-{\frac {\partial X}{\partial y}}.{\frac {\partial Y}{\partial x}}$$