OF ROTATING COMPRESSIBLE MASSES. 
197 
and, on inserting the value of x 2 ja 2 given by equation (167), this becomes 
2r 2u _ 2L /L 
a 4 a 2 a 4 
^+ 2 -? 
a a 
+ ... = 0 . 
42. Let us examine in particular the special form assumed by this equation for the 
special configuration at which the pseudo-spheroidal form becomes unstable, giving 
place to a pseudo-ellipsoidal form. Inserting the numerical values for L, p, It, &c., 
obtained in §§ 18-39, the equation becomes 
l+e[(y-2)-r0509]+e 2 [!(y-2) 2 -0-4063 (y-2)-0-0510] + ... = 0. . (168) 
For a given value of y this equation determines the value of e for which centri¬ 
fugal force just outbalances gravity as the pseudo-spheroidal form gives place to the 
pseudo-ellipsoidal. 
For instance, for the value y = 2, the equation becomes 
l-r0509e-0-0510e 2 -.... 
The values of e obtained by using terms as far as e and e respectively are 0'9516 
and 0'9112. The true root is perhaps somewhere near e = yV Thus a mass of 
rotating matter obeying Laplace’s law (y = 2) will throw off matter from its 
equator before reaching the ellipsoidal point of bifurcation if e > yo, or if <x is less 
than y^p. 
The root e = y 9 ^ agrees well with the corresponding quantity in the two-dimensional 
problem. For cylindrical masses obeying Laplace’s law (y = 2) the problem can be 
solved exactly and the root is found to be e = 1, giving <x = 0. # 
Equation (168) may more usefully he regarded as giving y when the value of e is 
assigned. The only value of e which is of any astronomical interest is the value 
e — 1, the value for a mass whose density reduces to zero at the boundary. Putting 
e = 1, we obtain an equation giving a critical value of y. 
From terms as far as e only, the value of y is clearly enough 
y = 2-0509. 
On including terms as far as y 2 , this value becomes 
y = 2-1521. 
These values of y appear to be converging to a limit; we cannot state it with 
great accuracy, but we shall perhaps not be far wrong if we assume it to be y = 2"2. 
Here again we may compare the problem with the simpler two-dimensional one in 
which, as follows from what has already been said, the root of F = 0, when e = 1, 
is y = 2 exactly. 
* ; Phil. Trans.,’ A, vol. 213, p. 471. 
VOL. CCXVIII. - A. 2 D 
