Page:Somerville Mechanism of the heavens.djvu/110

Pressure of a Particle moving on a curved Surface.

84. If the particle be moving on a curved surface, it exerts a pressure which the surface opposes with an equal and contrary pressure.

Demonstration,—For if F be the resulting force of the partial accelerating forces X, Y, Z, acting on the particle at $m$, it may be resolved into two forces, one in the direction of the tangent $mT$, and the other in the normal $mN$, fig. 12. The forces in the tangent have their full effect, and produce a change in the velocity of the particle, but those in the normal are destroyed by the resistance of the surface. If the particle were in equilibrio, the whole pressure would be that in the normal; but when the particle is in motion, the velocity in the tangent produces another pressure on the surface, in consequence of the continual effort the particle makes to fly off in the tangent. Hence when the particle is in motion, its whole pressure on the surface is the difference of these two pressures, which are both in the direction of the normal, but one tends to the interior of the surface and the other from it. The velocity in the tangent is variable in consequence of the accelerating forces X, Y, Z, and becomes uniform if we suppose them to cease.

Centrifugal Force.

85. When the particle is not urged by accelerating forces, its motion is owing to a primitive impulse, and is therefore uniform In this case X, Y, Z, are zero, the pressure then arising from the velocity only, tends to the exterior of the surface.

And as $v$ the velocity is constant, if $ds$ be the element of the curve described in the time$dt$, then

$ds=vdt{\text{whence}}dt={\frac {ds}{v}}\,$

therefore $ds$ is constant; and when this value of $dt$ is substituted in