262 The Configurations of Rotating Compressible Masses [ch. ix 
I have calculated the coefficients in this configuration as far as terms in 
(p 0 — er) 3 . The value of <w 3 is found to be given by 
2^ = 0-18712+ 0*06827 + [0-01602-007098 (#-2)] + ... 
(238-1), 
so that the first effect of compressibility is to increase the value of g> 3 /27T7 p. 
The intercept on the ¿r-axis (the radius of the new equator) is found to be 
given by * 
- 2 =1 + [0-54102 - 0-49950 (* - 2)] 
+ [0-74761 - 1-13574 (# - 2) + 0-83190 (* - 2) 3 ] + ... 
(238-2), 
so that for some values of k at least the effect of compressibility will be to 
draw the equator of the figure further away from the axis of rotation. 
This extension of the equator combined with the increase in the value of 
© 2 /27ryp increases the ratio of centrifugal force to gravity. 
The question arises whether centrifugal force can ever equal or outweigh 
gravity at the equator of the pseudo-spheroid. Calculation shews that the 
two forces become precisely equal when 
1 + [0-9990 (k — 2) — 10500] 
+ [0-4997 (* - 2) 2 + 0-07140 (* - 2) - 0-07998] + ... = 0 
(238-3). 
When k is given, this equation determines a value of ( p 0 — a)/p 0 , such that 
centrifugal force just balances gravity at the instant at which the pseudo- 
spheroidal form is giving place to the pseudo-ellipsoidal. 
Alternatively the equation determines a critical value of * when (p 0 — a-)/p 0 
is assigned. It is this latter use of the equation which is of primary interest 
to us, the important case being ( p 0 — a)/p 0 — 1, which represents a mass such 
as an actual star in which the density falls to zero at the surface. Putting 
(p 0 — <r)/p 0 — 1 in equation (238'3) and ignoring terms in [(p 0 — o-)/p 0 ] 2 and 
beyond, the solution of the equation is found to be 
* = 2-0501 (2384), 
while if we include the terms in. [(/? 0 — cr)/p 0 ] 2 and neglect those beyond, the 
solution is 
k = 2-1252 (238-5). 
We cannot state with great accuracy the value of k to which these values 
are converging, but there is not likely to be any very great error in taking it 
to be k = 2"2. Assuming this value, it appears that a mass of gas for which 
* See Problems of Cosmogony and Stellar Dynamics, equation (495).