OF ROTATING COMPRESSIBLE MASSES. 
195 
40. The rotation at this point of bifurcation is given by 
2 
—— = n + e An + e 2 Sn 
^7 rp 0 
= 0T8712 —0'04400 )-[0'01292 + 0-05495 (y-2)](^^). (163) 
V Po / V po / 
The first point of interest about this equation is that the term in (p 0 —<r) is 
independent of y. This must necessarily be the case, for we have seen that for an 
incompressible mass, regarded as a special case of the compressible mass that has 
been under discussion, y — 2 is infinite while p 0 —cr vanishes, and the product of the 
two remains finite. Thus for an incompressible mass (163) reduces to 
w 2 /2n r/3 0 = O'18712, 
which is the true value, but if there has been a term in (y —2) multiplying (p 0 — <r), 
equation (163) would have led to a wrong value. 
When y = 2, equation (163) reduces to 
u> 2 / 2 7rp 0 = 0T8712 —0'04400e—0'01292e 2 —_ 
Although we cannot be perfectly sure, the series is almost certainly convergent 
right up to the limiting case of <r = 0 or e = 1. In this case, it appears to converge 
to a limit of about w 2 /27rp Q — O' 120, but unfortunately it is impossible to evaluate the 
limit for other values of y. 
A more important question is the relation between w 2 and p at the point of 
bifurcation. 
We. have evaluated p as far as e 3 , but for comparison with the value of w given by 
equation (163), it will be enough to evaluate p as far as e 2 . Furthermore, to avoid 
a very complicated integration, we shall use Approximation B for the terms in e 2 , and 
so take 
P - po- 
-{po — <r) 
x v ,z 
2 + ;2 + — +e>] 
a b & 
x 2 , y 2 , 2 2 ' 2 ' 
a° + V + ~c- 
where ^ \ (y — 2'75), this being the approximation given by equation (84). By 
a simple projective transformation, it follows that the mean density is the same as 
that in a sphere in which the law of density is 
P = Pi)—(pv—o)l — 2 +e»y—;) • 
\a at 
As far as terms in e, the radius of this sphere is given by r 0 = (l— ^eri)a, and the 
mass taken as far as e 2 is 
