Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/45

21.

Let us again give to the symbols ${\textstyle p,}$ ${\textstyle q,}$ ${\textstyle E,}$ ${\textstyle F,}$ ${\textstyle G,}$ ${\textstyle \omega }$ the general meanings which were given to them above, and let us further suppose that the nature of the curved surface is defined in a similar way by two other variables, ${\textstyle p',}$ ${\textstyle q',}$ in which case the general linear element is expressed by

$${\sqrt {E'\,dp'^{2}+2F'\,dp'.dq'+G'\,dq'^{2}}}$$

Thus to any point whatever lying on the surface and defined by definite values of the variables ${\textstyle p,}$ ${\textstyle q}$ will correspond definite values of the variables ${\textstyle p',}$ ${\textstyle q',}$ which will therefore be functions of ${\textstyle p,}$ ${\textstyle q.}$ Let us suppose we obtain by differentiating them

$${\begin{alignedat}{2}dp'&=\alpha \,dp&&+\beta \,dq\\dq'&=\gamma \,dp&&+\delta \,dq\end{alignedat}}$$

We shall now investigate the geometric meaning of the coefficients ${\textstyle \alpha ,}$ ${\textstyle \beta ,}$ ${\textstyle \gamma ,}$ ${\textstyle \delta .}$

Now four systems of lines may thus be supposed to lie upon the curved surface, for which ${\textstyle p,}$ ${\textstyle q,}$ ${\textstyle p',}$ ${\textstyle q'}$ respectively are constants. If through the definite point to which correspond the values ${\textstyle p,}$ ${\textstyle q,}$ ${\textstyle p',}$ ${\textstyle q'}$ of the variables we suppose the four lines belonging to these different systems to be drawn, the elements of these lines, corresponding to the positive increments ${\textstyle dp,}$ ${\textstyle dq,}$ ${\textstyle dp',}$ ${\textstyle dq',}$ will be

$${\sqrt {E}}.dp,\quad {\sqrt {G}}.dq,\quad {\sqrt {E'}}.dp',\quad {\sqrt {G'}}.dq'.$$

The angles which the directions of these elements make with an arbitrary fixed direction we shall denote by ${\textstyle M,}$ ${\textstyle N,}$ ${\textstyle M',}$ ${\textstyle N',}$ measuring them in the sense in which the second is placed with respect to the first, so that ${\textstyle \sin(N-M)}$ is positive. Let us suppose (which is permissible) that the fourth is placed in the same sense with respect to the third, so that ${\textstyle \sin(N'-M')}$ also is positive. Having made these conventions, if we consider another point at an infinitely small distance from the first point, and to which correspond the values ${\textstyle p+dp,}$ ${\textstyle q+dq,}$ ${\textstyle p'+dp',}$ ${\textstyle q'+dq'}$ of the variables, we see without much difficulty that we shall have generally, i.e., independently of the values of the increments ${\textstyle dp,}$ ${\textstyle dq,}$ ${\textstyle dp',}$ ${\textstyle dq',}$

$${\sqrt {E}}.dp.\sin M+{\sqrt {G}}.dq.\sin N={\sqrt {E'}}.dp'.\sin M'+{\sqrt {G'}}.dq'.\sin N'$$

since each of these expressions is merely the distance of the new point from the line from which the angles of the directions begin. But we have, by the notation introduced above,

$$N-M=\omega .$$

In like manner we set

$$N'-M'=\omega ',$$