EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
393 
We attribute to Jc in (49) all values from oo to 2, and thus find a series of equations 
for the X’s. A similar series of equations holds for the A’s. 
We must now find the outstanding potential of the first degree of harmonics. No 
such term exists in (39), (41), (43), but it arises entirely out of (40') and (42'). If we 
write v 1 for the outstanding potential, we have clearly 
{W-Ydh +i(‘YT ; 
whence 
AttA'’’ 
5 ~' c] 2! 1 ! c a 
'A" 
c) i- 2 il 11 i — 1 c a 
a \ 3 1 6 
cl | \ a 
+‘t (i + D ,( : . . (so) 
i ~ 2 i — l J c a x ' 
and, by symmetry, 
5. Disturbance due to the Sectorial Harmonic Rotational Term. 
In (35) and (36) we have found this term to be — \oTq^ or — ^o) 2 Q 2 . 
d 2 d 2 
We have already observed that, if the operation — — — or S 2 be performed on w, 
CtOG Cblf 
the result vanishes when i— 1, 2, 3. 
Now, by (6), 
w * = i (-)Vn 
lc= 0 
4! 
7t'! 2 4 — 2k ! 
r-A — 2 IJc 
z p 
4 t 41 
- z 4 _ * — z 2 _]_0 
— " 1! 3 2! ^^2! 2 0!^' 
Hence \d 2 wf dp~ — 3, and, since Shtq = | (sr — if) d 2 wjdp 2 , it follows that 
q 2 = x 2 — if — and Q z = P' 3 W 4 . .(52) 
Hence the sectorial rotational term is — or 8 2 w i or — yj 00 2 S 2 ; this potential 
is of the second order of sectorial harmonics. 
Now, with e as defined in (37), let us assume as the equations to the two surfaces, 
r 
a 
A 2 
A\ 3; = 00 2 i + 1 
a) i = 2 27 — 2 
a 
A 
a 
i + 1 
8~W; 
mi 
1 + 2 
1 
%Ts ’tdfirA 
i = 2 27 
7 
S 2 W ; 
I + 2 
iX 
(53) 
We have now to determine the potentials of the inequalities on the two spheroids 
expressed by (53). 
MDCCCLXXXVII.—A. 3 E 
