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 $L,$ the two great circles drawn through its extremities and

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