OF ROTATING COMPRESSIBLE MASSES. 
177 
Clearly the error is quite large, but is concentrated in the coefficients L', m', n' 
which multiply terms in x. It is remarkable that the cross section of the figure by 
the plane x = 0 is given accurately to within 1 per cent, by this approximation. 
20. It appears that Approximation A will give a solution accurate to within 
2 per cent, for spheroidal figures, but of error varying from 2 to about 20 per 
cent, for ellipsoidal figures. A tolerable approximation to any required ellipsoidal 
figure could perhaps be got by regarding Approximation A as a first approximation, 
and obtaining the error in this by interpolation between the two errors which have 
been accurately estimated. 
21. A second approximate solution, which has an interest other than that of 
accuracy, may be referred to here. We may use the approximate equations (69) 
to simplify the approximate solution (68), and so obtain a still less accurate 
approximation, which we shall call Approximation B. The approximation is 
V = — — b 2 c 2 , &c. 
26 
(82) 
and the complete value of P 0 becomes 
p _ l 
•In — 9. 
(y-2) 
k 
e 
2 2 2 \ 2 
a 2 b 2 c 2 
(83) 
In this approximation the quantity k is at our choice; we must select it so as to 
give as good an average value as possible for the approximately equal quantities 
which occur in equations (69). 
Choosing a suitable value of k, I find for the coefficients at the ellipsoidal point of 
bifurcation 
L = M = n = B0273 (y-2)-07705 
l=m = 0-3488 (y-2) -0*2616 
> (Approximation B). . 
(84) 
N = 0-11845 (y —2) —0"0888 _ 
Comparing this with the exact solution given in equations (79), it appears that 
the error is as great as 20 per cent, when y = 2. 
The corresponding approximation at the pear-shaped point of bifurcation is almost 
worthless, the error being one of fully 50 per cent. 
The significance of these approximations will appear later. 
22. We turn now to the evaluation of the terms of second degree m x, y, z. 
It has already been seen (p. 172) that p" = q" = t" — 0, so that p, q, r aie identical 
with p', q', r f . Or, in other words, p, q , r do not involve (y — 2). 
Let us suppose that in the general ellipsoidal solution (cf. equation (/ 3)) 
f[p_l fDP + v\f 2 D 2 P]— = fourth degree terms + 4 (d v v 2 + cl 2 y 2 + d 3 z 3 ). . (85) 
Jo A 
