15'. Then the cubes of the
perpendiculars let fall from the eye, on the plane of the bases of the
various visual cones, are proportional to the solid contents of the
cones themselves, or, as the stars are supposed equally scattered within
all the cones, the cube roots of the numbers of stars in each of the
fields express the relative lengths of the perpendiculars. A _section_
of the sidereal system along any great circle can be constructed from
the data furnished by the gauges in the following way:

The solar system is within the mass of stars. From this point lines are
drawn along the different directions in which the gauging telescope was
pointed. On these lines are laid off lengths proportional to the cube
roots of the number of stars in each gauge. The irregular line joining
the terminal points will be approximately the bounding curve of the
stellar system in the great circle chosen. Within this line the space is
nearly uniformly filled with stars. Without it is empty space. A similar
section can be constructed in any other great circle, and a combination
of all such would give a representation of the shape of our stellar
system. The more numerous and careful the observations, the more
elaborate the representation, and the 863 gauges of HERSCHEL are
sufficient to mark out with great precision the main features of the
Milky Way, and even to indicate some of its chief irregularities.

On the fundamental assumption of HERSCHEL (equable distribution), no
other conclusion can be drawn from his statistics but the one laid down
by him.

This assumption he subsequently modified in some degree, and was led to
regard his gauges as indicating not so much the _depth of the system_ in
any direction, as the _clustering power or tendency_ of the stars in
those special regions. It is clear that if in any given part of the sky,
where, on the average, there are ten stars (say) to a field, we should
find a certain small portion having 100 or more to a field, then, on
HERSCHEL'S first hypothesis, rigorously interpreted, it would be
necessary to suppose a spike-shaped protuberance directed from the
earth, in order to explain the increased number of stars. If many such
places could be found, then the probability is great that this
explanation is wrong. We should more rationally suppose some real
inequality of star distribution here. It is, in fact, in just such
details that the method of HERSCHEL breaks down, and a careful
examination