OF ROTATING COMPRESSIBLE MASSES. 
179 
while the boundary is 
a* 
r 
b 2 
+ ¥, + 4=i-^^-(r-2)), 
2 p 0 \6 
(95) 
and so is still ellipsoidal. So far as this approximation goes, it appears that the 
shape of the ellipsoid, as given by the ratio of the semi-axes, depends only on 
o> 2 /27rp 0 ; it is the same as for a homogeneous mass of uniform density p 0 , equal to the 
density at the centre of the compressible mass, rotating with the same angular 
velocity. 
24. When this somewhat unsatisfactory approximation is abandoned there is no 
means of procedure except to calculate e„ e 2 and e 3 directly from equations (89), &c. 
Suppose that instead of being given by equation (33) the tidal potential had been 
given by 
V T + e A V x = (tj + cAtj) x 2 + (t 2 + £At.j) y 2 + (r 3 + eAr 3 ) 
■ (96) 
Then on equating terms independent of e in the principal equation we should obtain 
the same equation as before, but in place of equations (86) to (88), there would be 
three equations such as 
4d,-M = A „+ 
a 7rp 0 abc 
or, inserting the value of 4cZ, from equation (89), 
2 jpJaa ^2 ^aa — I2 Iab — ^2 Iac + 4^1 — ^4 — An + 
2p0 
At, 
(97) 
. _ 7 2 J 'A14 2 j 'AC • XV 1 4 -- 1 „T _ " 
a 0 c a irp^abc 
These equations become identical with equations (92) if we take 
At, = 4:7rp () abce 1 , &c.,.(98) 
and the equations then have the simple solution 
p = q = r = 0, An = 0. 
Thus the exact solution can be found by superposing the fourth degree terms 
already calculated on to an ellipsoid which is a figure of equilibrium under an 
additional tidal potential 
e AV T = i7rp 0 abce (e 1 x 2 + e 2 y 2 + e 3 z 2 ), .(99) 
this necessarily being harmonic, since e 1 + e 2 + e 3 = 0. 
25. Let a', b', c' be the semi-axes of a figure of equilibrium under this additional 
tidal force, these differing from the old quantities a, b, c by small quantities of the 
order of e. The equations determining a\ b\ c' are {of. equations (38) to (41)) 
/2 
- ttt ,; + — 
7 rp 0 CC b C 
co 
6' 
2-n-p (J a'b'c > a ' 
( 100 ) 
