Page:Somerville Mechanism of the heavens.djvu/111

equation (7), in consequence of the values of $X',Y',Z'$, in the end of article 69, it gives

$v^{2}.{\frac {d^{2}x}{ds^{2}}}=R_{'}\cos \alpha$

$v^{2}{\frac {d^{2}y}{ds^{2}}}=R_{'}\cos \beta$

$v^{2}.{\frac {d^{2}z}{ds^{2}}}=R_{'}\cos \gamma$

for by article 81 the particle may be considered as free, whence

$R_{'}={\frac {v^{2}{\sqrt {(d^{2}x)^{2}+(d^{2}y)^{2}+(d^{2}z)^{2})}}}{ds^{2}}}$;

and as the osculating radius is

$r={\frac {ds^{2}}{\sqrt {(d^{2}x)^{2}+(d^{2}y)^{2}+(d^{2}z)^{2}}}}$,

so

$R_{'}={\frac {v^{2}}{r}}$

The first member of this equation was shown to be the pressure of the particle on the surface, which thus appears to be equal to the square of the velocity, divided by the radius of curvature.

86. It is evident that when the particle moves on a surface of unequal curvature, the pressure must vary with the radius of curvature.

87. When the surface is a sphere, the particle will describe that great circle which passes through the primitive direction of its motion. In this case the circle $AmB$ is itself the path of the particle; and in every part of its motion, its pressure on the sphere is equal to the square of the velocity divided by the radius of the circle in which it moves; hence its pressure is constant.

88. Imagine the particle attached to the extremity of a thread assumed to be without mass, whereof the other extremity is fixed to the centre of the surface; it is clear that the pressure which the particle exerts against the circumference is equal to the tension of the thread, provided the particle be restrained in its motion by the thread alone. The effort made by the particle to stretch the thread, in order to get away from the centre, is the centrifugal force.D2