Page:Somerville Mechanism of the heavens.djvu/107

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

${\frac {dx.d^{2}x+dy.d^{2}y+dz.d^{2}z}{dt^{2}}}=Xdx+Ydy+Zdz+R_{'}\{dx.\cos \alpha +dy.\cos \beta +dz.\cos \gamma \}$

${\text{R}}_{'}$, 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

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

and

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

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.