EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
399 
l =1 + 4~ vHiJl !£(!)'>+*> 9 - «, * 
+ 2 
R 
A - 1 + C 
\ _ X~\ ^ {a_Y 1 y° 2z + l {Ay + 1 
Ii 2 ) ' \A 
i = 2 2i-2\c 
RJ 
y ( 72 ) 
From (27), (49), (65), we see that h 2 , h s . . . A,- . . ., X 2 , X 3 . . . X; . . /x 2 , p 3 
are to be found by solving the equations resulting from all values of & from 2 to 
infinity in the following :— 
,i —1 
*• - 1 =* & ft r > + «) 2 (?T (f)*! K ft <- r i B h > 
X/- — 1 
«.\ 3 [a \ 3 
(A + 1) (Jc + 2)\c ) \A 
1 
(Jc + 1) (k -f- 2) 
a 
A ft a, r] 
~c) i?2 ^ ^ ^ 
//v\3 / J\3i = 
ar h ' ' ' 
a\ 5 / 
/ /y \ 3 / yd \ 3 i — 
3 \2 / _ \ / \ s 
/«-1 = l(r) (7) R r ) + ft) a (“) (7),., 
= 00 
y 
/ft i, r y 1 (ii 
(73) 
and symmetrical systems of equations for obtaining the i/’s, As, and M’s. 
With the values found by the solution of these equations we then evaluate K by 
formula (71); and we have 
We are now enabled to find the Vs and m’s by the formulte (48) and (64), viz., 
h — 1 V e (^ + 1) (& + 2) (^j h 
, f A \* 
m i = TO e (7 ) H 
and the symmetrical forms give us the L ’s and M’s. 
Having thus evaluated all the auxiliary constants, (72) gives the solution of the 
problem. 
It is well known that f X 3m 3 /47r is the ellipticity of a single homogeneous mass of 
fluid rotating with angular velocity a>. Hence the first terms of (72) simply denote 
the ellipticity due to rotation in each of the masses, as if the other did not exist. 
Now the rigorous solution for the form of equilibrium of a rotating mass of fluid is an 
ellipsoid of revolution with eccentricity sin g, the value of g being given by the 
solution of 
(O^ 1 
~ = cot 3 g [(3 -f tan 3 g) g - 3 tan g] *.(76) 
* See, for example, Thomson and Tait’s ‘ Natural Philosophy ’ (3), § 771, with / = tang. 
