Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/97

may then be called its mean curvature. Curvature, on the contrary, always presupposes that the point is determined, and is defined as the mean curvature of an element at this point; it is therefore equal to

$${\frac {d\Lambda }{d\,al}}.$$

We see, therefore, that arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density. The reciprocal of the curvature, namely,

$${\frac {d\,al}{d\Lambda }},$$

is called the radius of curvature at the point ${\textstyle l.}$ And, in keeping with the above conventions, the curve at this point is called concave toward the right and convex toward the left, if the value of the curvature or of the radius of curvature happens to be positive; but, if it happens to be negative, the contrary is true.

If we refer the position of a point in the plane to two perpendicular axes of coordinates to which correspond the directions ${\textstyle 0}$ and ${\textstyle 90^{\circ },}$ in such a manner that the first coordinate represents the distance of the point from the second axis, measured in the direction of the first axis; whereas the second coordinate represents the distance from the first axis, measured in the direction of the second axis; if, further, the indeterminates ${\textstyle x,}$ ${\textstyle y}$ represent the coordinates of a point on the curved line, ${\textstyle s}$ the length of the line measured from an arbitrary origin to this point, ${\textstyle \phi }$ the direction of the tangent at this point, and ${\textstyle r}$ the radius of curvature; then we shall have

$${\begin{aligned}dx&=\cos \phi .ds,\\dy&=\sin \phi .ds,\\r&={\frac {ds}{d\phi }}.\end{aligned}}$$

If the nature of the curved line is defined by the equation ${\textstyle V=0,}$ where ${\textstyle V}$ is a function of ${\textstyle x,}$ ${\textstyle y,}$ and if we set

$$dV=p\,dx+q\,dy,$$

then on the curved line

$$p\,dx+q\,dy=0.$$

Hence

$$p\cos \phi +q\sin \phi =0,$$