OF ROTATING COMPRESSIBLE MASSES. 
205 
in which e must of course be put equal to unity for a gas in which <r = 0 at the 
boundary. The series on the right is probably rapidly convergent, but sufficient 
terms have not been calculated for the value of y to be determined with accuracy. 
Neglecting all terms beyond those written down, the root of equation (182), when 
e = 1, is found to be y = 2'1521, but the remaining terms appear likely to increase 
this value somewhat, and we may perhaps take y = 2'2 as an approximate value. 
This, we may notice, is just half-way between the two guess-values considered in § 50. 
When the density at the boundary is not zero, the critical value of y is less than 
this; for instance, a critical value y = 1 corresponds to a value of e equal to about 
three-quarters. 
An approximately accurate drawing of the critical figure, when e = 1, is shown in 
fig. 4, the inner curves being strata of constant density. 
51. We have considered the effect of heterogeneity in the structure of the matter, 
and have found that a sinking of the heavier elements to the centre of the mass will 
result in an increase in the critical value of y. There is no limit to the amount of 
increase that can be produced in this way, although naturally the amount of increase 
depends on the extent to which the light and heavy elements are separated and on 
the ratio of their amounts. As an illustration, we considered the extreme case of a 
core of heavy incompressible material which we called iron, surrounded by an 
atmosphere of much lighter material which we called hydrogen, the two elements 
being supposed to be completely separated. If the volume of hydrogen was initially 
greater than about one-third of that of iron, the composite mass set into rotation will 
first disintegrate through equatorial loss of matter. 
A drawing of the critical figure has already been shown on fig. 3 (p. 200); this 
may be compared with fig. 4. 
52. There is, however, a very essential difference between the failure of a uniform 
mass to attain the ellipsoidal form and the corresponding failure of a heterogeneous 
mass. Briefly speaking the former is permanent, while the latter is transitory. 
Consider for simplicity a mass of perfectly uniform gas—say helium for which 
y = if—set into rotation and continually shrinking by cooling. The value of y is 
VOL. ccxviii.—a. 2 E 
