HERSCHEL endeavored, in his early memoirs, to explain this inequality of distribution on the fundamental assumption that the stars were nearly equably distributed in space. If they were so distributed, then the number of stars visible in any gauge would show the thickness of the stellar system in the direction in which the telescope was pointed. At each pointing, the field of view of the instrument includes all the visible stars situated within a cone, having its vortex at the observer's eye, and its base at the very limits of the system, the angle of the cone (at the eye) being 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 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.