EQUILIBRIUM OF ROTATING MASSES OF FLUID. 419 
Jacobi’s Ellipsoids. 
Axes. 
Ang. vel. 
Mom. 
Greatest 
Mean 
Least 
, .2 
a 
b 
c 
[1 
b 
b 
b 
4w 
1 
1 1972 
1-1972 
•6977 
•09356 
•30375 
2 
U279 
1-123 
•696 
•093 
•306 
3 
1-383 
1-045 
•692 
•0906 
•3134 
4 
I -601 
•924 
■677 
•0830 
•3407 
5 
1-899 
•811 
•649 
•0705 
•3920 
6 
2-346 
•702 
•607 
•0536 
•4809 
7 
3-136 
•586 
•545 
•0334 
- 644 
8 
5-04 
•45 
•44 
•013 
1-016 
9 
OO 
•00 
•00 
•000 
OO 
In figs. 4 and 5, Plates 22 , 23, w s / 47 t is ‘038, and the momentum p, is ’472. On com¬ 
parison with the Table of Jacobi’s ellipsoids, we see that this corresponds with a con¬ 
siderably slower rotation than the 6 th solution, and nearly the same moment of 
momentum. 
In the next case the two masses are still closer (/3 = 3 -), the distance between the 
centres being only 2 '449 times either mean radius. The result is illustrated in 
figs. 2 and 3 ; the explanation of figs. 4 and 5 serves, mutatis mutandis, for these 
two figures also. 
This case is interesting because the masses have approached so near to one another 
that they partially overlap. Two portions of matter cannot, of course, occupy the 
same space, and the continuity of figures of equilibrium leads us to believe that the 
reality must consist of a single mass of fluid. In figs. 2 and 3 conjectural dotted 
lines are drawn to show how it is probable that the overlapping of the two masses is 
replaced by a neck of fluid joining them. The figures as thus amended serve to give 
a good representation of the single dumb-bell shaped figure of equilibrium. 
The angular velocity is here given by or jin = ‘049, and the moment of momentum 
by - 482. In the sixth entry of the Table of Solutions of Jacobi’s problem we find 
co' 3 / 47 t = ‘0536, and the moment of momentum p = ‘481. This ellipsoid has, then, the 
same moment of momentum, and only about 4 per cent, more angular velocity, than 
our dumb-bell. It has seemed, therefore, worth while to mark (in chain-dot) on 
figs. 2 and 3 the outline of this Jacobian ellipsoid of the same mass as the dumb¬ 
bell. The actual vertex of the ellipsoid just falls outside the limits to which it was 
possible to extend the figure. 
In the paper above referred to it is shown how the energy of the Jacobian ellipsoid 
is to be computed. If we denote the kinetic energy by (fTr^b 5 X e, and the intrinsic 
energy by ( 377-) 2 b 5 X (i — 1),* then it appears that in the case of the ellipsoid drawn 
in these figures e = ‘0964, i = ’4808, and the total energy E = e + i — ‘5772. 
# The intrinsic energy being negative, it is more convenient to tabulate i a positive quantity. 
3 H 2 
