FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 19 
must cover all possible figures of equilibrium as far as the third order. One such 
figure is, however, known to be 
0 = e. 0 + e 2 .0 + e 3 P,.(85) 
and the corresponding solution for p, q, r is that expressed by equations (83). 
From equations (82), it follows at once that we must have 
/. v y 
A-i, ft. i, ft- 1 
h, h',, h" 2 
i h, K 
Indeed, it is now clear that this is precisely the equation which determined the 
existence of the point of bifurcation on the series of Jacobian ellipsoids.* 
If this were the only relation between the coefficients in equations (80), these 
equations could have no solution other than p = q = !' = oo. Let us, however, multiply 
the three equations (80) by the three minors of k'\, k" 2 , k " 3 in the determinant of 
equation (86), and add. We obtain a relation of the type, 
Jc l , k\, 
p.O + q.O + r.O = h, k' 2 , % 
K k' 3 , % 
(87) 
and it is clear that these equations can now have a solution in which p, q, r are not all 
infinite if we have 
k 2 ^ 
h, 
k\, 
k' 2 , 
k'. 
3 ) 
% 
= 0 
( 88 ) 
It is only when this relation is satisfied that it is possible to continue our linear 
series of equilibrium configurations beyond the second order terms. When it 
is not satisfied, our third order solution (84) lapses back into the solution (85), 
namely <p = e 3 P, which is virtually the first order solution with e 3 replacing e as 
parameter. 
16. The relation (88) can be expressed in a much simpler form. 
Independently of the values of p, q and r, we have already seen (equation (71)) 
that we must have 
Stq + tq T — 0.. 
* As to the relation of this equation to the general theory, see § 36 of my previous paper. 
D 2 
(89) 
