Page:The Solar System - Six Lectures - Lowell.djvu/60

42

The Solar System

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Of this, the first term is the potential of gravity; the subsequent ones the tide-raising potential.

To get the forces, we must differentiate this expression with regard to the position of the particle.

Tide-rising force for different planets.

In order to compare the tide-raising forces on different different bodies, we will assume $z=0$ whence the tide-raising force at its maximum may be expressed in a rapidly converging series, of which the first two terms are $\textstyle {\frac {2m\rho }{r^{3}}}+{\frac {3m\rho ^{2}}{r^{4}}}.$

If the affected body be distant compared with its size, the first term is enough, and we see that then the tide-raising force is directly as the radius of the second body, and inversely as the cube of its distance from the first, while also directly as the latter's mass.

But the work done by a force is the product of the force into the space through which it acts,—as, for instance, the lifting a weight a certain distance,—and in a given time the space is itself proportional to the force, whence the work in that time is as the square of the force.

$fdt=dv={\frac {ds}{dt}},$

whence

${\frac {1}{2}}ft^{2}=s$

Whence if the time remain constant the force must vary as the space. For the proportionate work