centre of curvature N, which is the intersection of two consecutive normals mN, m'N, never varies in the circle and sphere, because the curvature is every where the same; but in all other curves and surfaces the position of N changes with every point in the curve or surface, and a, b, c, are only constant from one point to another. By this property, the equation of the radius of curvature is formed from the equation of the curve, or surface. If r be the radius of curvature, it is evident, that though it may vary from one point to another, it is constant for any one point m where δr=0.
Equilibrium of a Particle on a curved Surface.
49. The equation (3) is sufficient for the equilibrium of a particle of matter, if it be free to move in any direction; but if it be constrained to remain on a curved surface, the resulting force of all the forces acting upon it must be perpendicular to the surface, otherwise it would slide along it; but as by experience it is found that reaction is equal and contrary to action, the perpendicular force will be resisted by the re-action of the surface, so that the re-action is equal, and contrary to the force destroyed; hence if R′ be the resistance of the surface, the equation of equilibrium will be
Xδx+Yδy+Zδz=-
δx, δy, δz are arbitrary; these variations may therefore be assumed to take place in the direction of the curved surface on which the particle moves; then by the property of the normal, δr=0; which reduces the preceding equation to
Xδx+Yδy+Zδz=0.
But this equation is no longer equivalent to three equations, but to two only, since one of the elements δx, δy, δz, must be eliminated by the equation of the surface.
49. The same result may be obtained in another way. For if u=0 be the equation of the surface, then δu=0; but as the equation of the normal is derived from that of the surface, the equation δr=0 is connected with the preceding, so that δr=Nδx. But
whence
