Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/21

 which is infinitely small; and finally, let $\lambda$ be the point on the sphere representing the direction of the element $AA'.$  Then we shall have

$$dx = ds. \cos (1)\lambda,\quad dy = ds. \cos (2)\lambda,\quad dz = ds. \cos (3)\lambda$$

and, since $\lambda L$  must be equal to $90^{\circ},$

$$X\cos (1)\lambda + Y\cos (2)\lambda + Z\cos (3)\lambda = 0$$

By combining these equations we obtain

$$X\, dx + Y\, dy + Z\, dz = 0.$$

There are two general methods for defining the nature of a curved surface. The first uses the equation between the coordinates $x,$  $y,$  $z,$ which we may suppose reduced to the form $W=0,$  where $W$  will be a function of the indeterminates $x,$  $y,$  $z.$  Let the complete differential of the function $W$  be

$$dW = P\, dx + Q\, dy + R\, dz$$

and on the curved surface we shall have

$$P\, dx + Q\, dy + R\, dz = 0$$

and consequently,

$$P \cos (1)\lambda + Q \cos (2)\lambda + R \cos (3)\lambda = 0$$

Since this equation, as well as the one we have established above, must be true for the directions of all elements $ds$ on the curved surface, we easily see that $X,$  $Y,$  $Z$  must be proportional to $P,$  $Q,$  $R$  respectively, and consequently, since

$$X^{2} + Y^{2} + Z^{2} = 1,$$

we shall have either

$$\begin{aligned} X &= \frac{P}{\sqrt{P^{2} + Q^{2} + R^{2}}}, & Y &= \frac{Q}{\sqrt{P^{2} + Q^{2} + R^{2}}}, & Z &= \frac{R}{\sqrt{P^{2} + Q^{2} + R^{2}}} \end{aligned}$$

or

$$\begin{aligned} X &= \frac{-P}{\sqrt{P^{2} + Q^{2} + R^{2}}}, & Y &= \frac{-Q}{\sqrt{P^{2} + Q^{2} + R^{2}}}, & Z &= \frac{-R}{\sqrt{P^{2} + Q^{2} + R^{2}}} \end{aligned}$$

The second method expresses the coordinates in the form of functions of two variables, $p,$  $q.$ Suppose that differentiation of these functions gives

$$\begin{alignedat}{2} dx &= a\, dp &&+ a'\, dq \\ dy &= b\, dp &&+ b'\, dq \\ dz &= c\, dp &&+ c'\, dq \end{alignedat}$$