Page:Somerville Mechanism of the heavens.djvu/107

Chap II.] variations be changed into differentials, and if $$X', Y', Z'$$ be eliminated by their values in the end of article 69, that equation becomes

$\frac{dx.d^2x+dy.d^2y+dz.d^2z}{dt^2}=Xdx+Ydy+Zdz

+R_'\{dx.\cos\alpha+dy.\cos\beta+dz.\cos\gamma\}$

$$\text{R}_'$$, being the reaction in the normal, and $$\alpha,\beta,\gamma$$ the angles made by the normal with the co-ordinates. But the equation of the surface being $$u = 0$$,

$du=\frac{du}{dx}.dx+\frac{du}{dy}.dy+\frac{du}{dz}.dz = 0$;

consequently, by article 69,

$\lambda du = dx .\cos\alpha + dy. \cos\beta + dz.\cos\gamma =0$;

so that the pressure vanishes from the preceding equation; and when the forces are functions of the distance, the integral is

$2f(x,y,z) +c = v^2$

$A^2-v^2=f(x,y,z)-2f(a,b,c),$

as before. Hence, if the particle be urged by accelerating forces, the velocity is independent of the curve or surface on which the particle moves; and if it be not urged by accelerating forces, the velocity is constant. Thus the principle of Least Action not only holds with regard to the curves which a particle describes in space, but also for those it traces when constrained to move on a surface.

82. It is easy to see that the velocity must be constant, because a particle moving on a curve or surface only loses an indefinitely small part of its velocity of the second order in passing from one indefinitely small plane of a surface or side of a curve to the consecutive; for if the particle be moving on $$ab$$ with the velocity $$v$$; then if the angle $$abe = \beta$$, the velocity in $$bc$$ will be $$v\cos\beta$$; but $$\cos\beta =1 - \frac{1}{2}\beta^2-$$ &c.; therefore the velocity on $$bc$$ differs from the velocity on $$ab$$ by the indefinitely small quantity $$\frac{1}{2} v. \beta^2$$. In order to determine the pressure of the particle on the surface, the analytical expression of the radius of curvature must be found.