Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/115

 Hence we have

$$d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta - (Y\xi - X\eta)\, d\zeta + \xi\zeta\, dY - \eta\zeta\, dX} {(Y\xi - X\eta)^{2} + \zeta^{2}}.$$

The denominator of this expression is

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

or

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

We verify readily by expansion the identical equation

$$\begin{array}{c} \eta\zeta(X^{2} + Y^{2} + Z^{2}) + YZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\ = (X\xi + Y\eta + Z\zeta)(Z\eta + Y\zeta) + (X\zeta - Z\xi)(X\eta - Y\xi) \end{array}$$

and likewise

$$\begin{array}{c} \xi\zeta(X^{2} + Y^{2} + Z^{2}) + XZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\ = (X\xi + Y\eta + Z\zeta)(X\zeta + Z\xi) + (Y\xi - X\eta)(Y\zeta - Z\eta). \end{array}$$

We have, therefore,

$$\begin{aligned} \eta\zeta &= -YZ + (X\zeta - Z\xi )(X\eta - Y\xi), \\ \xi\zeta &= -XZ + (Y\xi   - X\eta)(Y\zeta - Z\eta). \end{aligned}$$

Substituting these values, we obtain

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

$$ + \frac{X\eta - Y\xi}{1 - Z^{2}}\bigl\{ d\zeta - (X\zeta - Z\xi)\, dX - (Y\zeta - Z\eta)\, dY\bigr\}. $$

Now

$$\begin{alignedat}{4} & X\, dX &&+    Y\, dY &&+     Z\, dZ &&= 0, \\ &\xi\, dX &&+ \eta\, dY &&+ \zeta\, dZ &&= -X\, d\xi - Y\, d\eta - Z\, d\zeta. \end{alignedat}$$

On substituting we obtain, instead of what stands in the parenthesis,

$$d\zeta - Z(X\, d\xi + Y\, d\eta + Z\, d\zeta).$$

Hence

$$ d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)+ \frac{d\xi}{1 - Z^{2}}\{\zeta Y - \eta X^{2}Z + \xi XYZ\} $$

$$ \begin{aligned} &- \frac{d\eta}{1 - Z^{2}}\{\zeta X + \eta XYZ - \xi Y^{2}Z\}\\ &+ d\zeta(\eta X - \xi Y). \end{aligned} $$