Page:Somerville Mechanism of the heavens.djvu/107

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Chap II.]
VARIABLE MOTION.
31

variations be changed into differentials, and if be eliminated by their values in the end of article 69, that equation becomes

, being the reaction in the normal, and the angles made by the normal with the co-ordinates. But the equation of the surface being ,

;

consequently, by article 69,

;

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

and

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 with the velocity ; then if the angle , the velocity in will be ; but &c.; therefore the velocity on differs from the velocity on by the indefinitely small quantity . In order to determine the pressure of the particle on the surface, the analytical expression of the radius of curvature must be found.