Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/116

Since, further,

$${\begin{aligned}\eta X^{2}Z-\xi XYZ&=\eta X^{2}Z+\eta Y^{2}Z+\zeta ZYZ\\&=\eta Z(1-Z^{2})+\zeta YZ^{2},\\\eta XYZ-\xi Y^{2}Z&=-\xi X^{2}Z-\zeta XZ^{2}-\xi Y^{2}Z\\&=-\xi Z(1-Z^{2})-\zeta XZ^{2},\end{aligned}}$$

our whole expression becomes

$${\begin{aligned}d\phi &={\frac {Z}{1-Z^{2}}}(Y\,dX-X\,dY)\\&\quad +(\zeta Y-\eta Z)\,d\xi +(\xi Z-\zeta X)\,d\eta +(\eta X-\xi Y)\,d\zeta .\end{aligned}}$$

15.

The formula just found is true in general, whatever be the nature of the curve. But if this be a shortest line, then it is clear that the last three terms destroy each other, and consequently

$$d\phi =-{\frac {Z}{1-Z^{2}}}(X\,dY-Y\,dX).$$

But we see at once that

$${\frac {Z}{1-Z^{2}}}(X\,dY-Y\,dX)$$

is nothing but the area of the part of the auxiliary sphere, which is formed between the element of the line ${\textstyle L,}$ the two great circles drawn through its extremities and 

${\textstyle (3),}$ and the element thus intercepted on the great circle through ${\textstyle (1)}$ and ${\textstyle (2).}$ This surface is considered positive, if ${\textstyle L}$ and ${\textstyle (3)}$ lie on the same side of ${\textstyle (1)\ (2),}$ and if the