of his system leads to the belief that it must be greatly
modified to cover all the known facts, while it undoubtedly has, in the
main, a strong basis.

The stars are certainly not uniformly distributed, and any general
theory of the sidereal system must take into account the varied tendency
to aggregation in various parts of the sky.

In 1817, HERSCHEL published an important memoir on the same subject, in
which his first method was largely modified, though not abandoned. Its
fundamental principle was stated by him as follows:

"It is evident that we cannot mean to affirm that the stars of the
fifth, sixth, and seventh magnitudes are really smaller than those
of the first, second, or third, and that we must ascribe the cause
of the difference in the apparent magnitudes of the stars to a
difference in their relative distances from us. On account of the
great number of stars in each class, we must also allow that the
stars of each succeeding magnitude, beginning with the first, are,
one with another, further from us than those of the magnitude
immediately preceding. The relative magnitudes give only relative
distances, and can afford no information as to the real distances at
which the stars are placed.

"A standard of reference for the arrangement of the stars may be had
by comparing their distribution to a certain properly modified
equality of scattering. The equality which I propose does not
require that the stars should be at equal distances from each other,
nor is it necessary that all those of the same nominal magnitude
should be equally distant from us."

It consisted in allotting a certain equal portion of space to every
star, so that, on the whole, each equal portion of space within the
stellar system contains an equal number of stars. The space about each
star can be considered spherical. Suppose such a sphere to surround our
own sun. Its radius will not differ greatly from the distance of the
nearest fixed star, and this is taken as the unit of distance.

Suppose a series of larger spheres, all drawn around our sun as a
centre, and having the radii 3, 5, 7, 9, etc. The contents of the
spheres being as the cubes of their diameters, the first sphere will
have 3 x 3 x 3 = 27 times the volume of the unit sphere, and will
therefore be large enough to contain 27 stars; the second will have 125
times the volume, and will therefore contain 125 s