428 
ON FIGURES'OF EQUILIBRIUM OF ROTATING MASSES. OF FLUID. 
.1, ■ 
o, 
o, 
n±N 2 , 
n 4 N 3 , 
. . . . | 
%zv 4 , • • 
0, 
1, 
o, 
^3 ^2’ 
m 3 A t 3 , 
%N 4 , .. 
0, 
0, 
1, 
^2’ 
n. 2 N 3 , 
« 2 n 4 , .. 
• •) 2^4’ 
NjjWg, 
l, 
o, 
0, 
. ., N.pn v 
N s n s , 
N 3 n 2 , 
o, 
1, 
0, 
■ N 4 n 4 , 
o, 
o, 
1, 
It will be observed that, by tbe notation (xi.), and tbe appropriate division of the columns and rows of 
the tidal determinant, it has been converted into a determinant in which the diagonal consists of ones. 
If we drop certain factors common to the whole, we have, by differentiating (viii.), 
d 2 E _ _ 3 /A\ 3 7 *-i d 2 E _ 3 faf r *" 1 ■ d 2 E _ d 2 E 
dn k 2 2 \cj ft — 1 ’ dN/? 2 \c / ft - 1 ’ dn k dm ~ dN k dN t 
d 2 E _ 7 *-i U-! Je + il 
dn k dNi 2 \cj \c) ft-I i- 1 k ! i ! ’ 
d~E _ _ 3 /A\* k + 2 ! 7 f -i c7 3 U _ , /o\® ft + 2! r*" 1 
dp k 2 2 \c) 2.ft-2! ft- l’ cAP* 2 2 W 2.ft-2! ft-I* 
<PP _ r3 x 2 M\ 3 M 3 A-+ 2! 7 *"i U" 1 ft + f! 
dp k dPi icj \c/ 2. ft — 2 ! 7c - 1 ’ i — 1 i - 21 ft + 2 ! ’ 
_ I a\* L U-i 7c + i! 
c/ \c/ 2 A - I t - i i -2! ft-2 ! .. 
cPU _ tPP _ c7 3 P _ rf 2 P _ Q 
dv k dpi dn k dPi dN k dPi dN k dpi 
d 2 E _ d 2 E 
dp k dpi dP k dP 
(xiii.) 
With these values (xiii.) we easily find, by substitution in (xi.), that 
n k Ni 
3 (A \ ? /a\i [ 7 *“ 1 r*“I "I * ft + t! 
2 [cj [cj |_ft - 1 i - lj ft! t! 
(xiv.) 
[This expression gives the value of each of the entries in the infinite determinant (xii.). 
Now it is possible, by a certain laborious investigation which I do not here reproduce, to develope 
this infinite “tidal” determinant in the form of a series, and then to show that, however close the two 
masses may be to one another, the series arising from the tidal determinant can never vanish. It may 
also be proved that the other infinite determinant, which results from “rotational” terms, is necessarily 
greater than the tidal determinant, and a fortiori can never vanish.]* 
Thus it might be held that stability must subsist in the figures of equilibrium until the two masses 
come into contact. But, as appears from § II, it is certain that, if one of the masses be smaller than the 
other, this cannot be the case. In fact, the investigation must break down on account of the imperfection 
arising from the use of spherical harmonic analysis. 
We have seen that the infinite determinant, which gives the coefficients of stability, splits into two 
parts when we rely on spherical harmonic analysis; but when instability ensues it must be brought 
about by the joint action of the tidal and rotational forces. It appears certain then that, if a rigorous 
analysis were.used, this separation would not take place. 
Although the present investigation proves thus to be abortive, I have thought it worthwhile to sketch 
the process, because it almost certainly indicates the line that must be pursued whenever a more 
rigorous analysis shall be applied to this difficult problem. 
* This paragraph, replacing the investigation referred to, added July 12, 1887. 
