Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/100

 of curvature becomes infinite or the curvature vanishes. Then, generally speaking, since here

$$-p\sin\phi + q\cos\phi$$

will change its sign, we have here a point of inflexion.

5.

The case where the nature of the curve is expressed by setting $y$  equal to a given function of $x,$ namely, $y = X,$  is included in the foregoing, if we set

$$V = X - y.$$

If we put

$$dX = X'\, dx,\qquad dX' = X''\, dx,$$

then we have

$$\begin{array}{c} p = X',\qquad q = -1, \\ P = X'', \qquad Q = 0,\qquad R = 0, \end{array}$$

therefore

$$\pm\frac{1}{r} = \frac{X''}{(1 + X'^{2})^{3/2}}.$$

Since $q$  is negative here, the upper sign holds for increasing values of $x.$ We can therefore say, briefly, that for a positive $X$  the curve is concave toward the same side toward which the $y$ -axis lies with reference to the $x$ -axis; while for a negative $X$  the curve is convex toward this side.

6.

If we regard $x,$  $y$ as functions of $s,$  these formulæ become still more elegant. Let us set

$$\begin{alignedat}{2} \frac{dx}{ds} &= x',\qquad& \frac{dx'}{ds} &= x'', \\ \frac{dy}{ds} &= y',\qquad& \frac{dy'}{ds} &= y''. \end{alignedat}$$

Then we shall have

$$\begin{alignedat}{2} x' &= \cos\phi,\qquad & y' &= \sin\phi, \\ x &= -\frac{\sin\phi}{r},\qquad & y &= \frac{\cos\phi}{r}; \end{alignedat} $$

or

$$ \begin{alignedat}{2} y' &= -rx,\qquad& x' &= ry, \end{alignedat}$$