OF ROTATING COMPRESSIBLE MASSES. 
173 
If these relations were strictly true, the value of P 0 ' would he 
so that, from the consideration of § 14, we should have 
f [F—J/DP'+sL/ 8 D 2 P']^ = 0. 
(70) 
(71) 
Thus the error is of the order of the error of the approximation (69) multiplied by 
the coefficients of the integral (71), which coefficients are found to be small. It may 
be noticed that equations (69) are strictly true in the spherical configuration in 
which a — b = c, and the error increases as a, b, c become more unequal. 
Thus the error in solution (68) is nil in the spherical configuration ; we shall now 
evaluate it exactly in two other configurations : (i.) the point of bifurcation of the 
spheroidal and ellipsoidal series, and (ii.) the point of bifurcation of the ellipsoidal 
and pear-shaped series. 
17. Ellipsoidal Point of Bifurcation .—At the point of bifurcation of the series of 
spheroids with the series of ellipsoids, the semi-axes a , b, c, and the values of n and 6 
are given by 
a = b = 1‘1972, c = (T69766, 
n = = 0-18712, 0 = 0-47126, 
2 7rpo 
the scale of length, which is entirely at our disposal, being chosen so as to make 
abc = ]. 
Since the configuration under discussion is now spheroidal, we have 
L = M = n, l — m, p — q, 
so that there are only three coefficients, say, L, m and N, to the terms of fourth 
degree, and two coefficients jo, r to terms of second degree. 
For any configuration, formula (48) expresses the internal potential of a solid 
of uniform unit density, so that 
V 2 [V.(1) +eAV ; (l) + e (1V ; - (1) + ...] = -4tt.(72) 
It follows that AV t (l), dV^l), &c., are all spherical harmonics. 
In the present problem in which a ~ b, these harmonics are also symmetrical about 
the axis of 2 , and so are functions of x 2 + y 2 and of 2 only, so that we may assume 
f [P-i/DF + J T /’D a P]A 
Jo Za 
.. AV,(1) 
it abc 
= 4c n [(ai 2 + ?/ 2 ) 2 —8 (x' 2 + /) 2 2 + fz 4 ] +4 : d l (x 2 + y 2 — 2s?), . • (73) 
2 A 
VOL. CCXVIII.-A. 
