path is only double as large, the time taken to describe that path is
the square root of eight times as great; in fact, the velocity of the
outer body will be only the square root of twice that of the inner
one. As, however, its distance from the sun is twice as great, it
follows that the moment of momentum of the outer body will be the
square root of twice that of the inner body. We may state this result
a little more generally as follows--

In comparing the moments of momentum of the several planets which
revolve around the sun, that of each planet is proportional to the
product of its mass with the square root of its distance from the
sun.

Let us now compare two spheres together, the diameter of one sphere
being double that of the other, while the times of rotation of the two
are identical. And let us now compare together the moments of momentum
in these two cases. It can be shown by reasoning, into which I need
not now enter, that the moment of momentum of the large sphere will be
thirty-two times that of the small one. In general we may state that
if a sphere of homogeneous material be rotating about an axis, its
moment of momentum is to be expressed by the product of its angular
velocity by the fifth power of its radius.

We can now take stock, as it were, of the constituents of moments of
momentum in our system. We may omit the satellites for the present,
while such unsubstantial bodies as comets and such small bodies as
meteors need not concern us. The present investment of the moment of
momentum of our system is to be found by multiplying the mass of each
planet by the square root of its distance from the sun; these products
for all the several planets form the total revolutional moment of
momentum. The remainder of the investment is in rotational moment of
momentum, the collective amount of which is to be estimated by
multiplying the angular velocity of each planet into its density, and
the fifth power of its radius if the planet be regarded as
homogeneous, or into such other power as may be necessary when the
planet is not homogeneous. Indeed, as the denser parts of the planet
necessarily lie in its interior, and have therefore neither the
velocity nor the radius of the more superficial portions, it seems
necessary to admit that the moments of momentum of the planets will be
proportional to some lower power of the radius than the fifth. The
total moment of momentum of the planets by rotation, when multi