EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
389 
We must now go back and determine the value of the outstanding potential of the 
first degree of harmonics, which will be annulled when rotation is imposed on the 
system. The potential is given in (22—i.) and (22—ii.); (22—i.) contributes nothing, and 
(22—ii.) gives us, for 1: — 1, 
4ttM 3 
3c 
-4 1 
a\ 4i = ”i+l! E'- 1 ' 
1! if i — 1 111 
i = 2 
W, 
a 
Thus, if we call U l the outstanding potential of the first degree, when referred to 
the two origins respectively, we have 
_ dTrM 3 f 3 
Ul ~ 3c [_ + 2 
q\ 3 «V>* + 1 r ,_ 
c) i=2 i-l 
l H; 
U\ —- 
47TC 3 
~37 
1+f 
-4\ Si =* i + 1 i 7 
7 S 
G/ j = — t 
a vj 
c a 
A _ W l 
c 
>• 
A 
. . (32) 
§ 3. The Potential due to Rotation. 
Intermediate between the two origins o and 0 take a third Q, and take the axes of 
£and 7] parallel to those of x and y, and that of £ identical with that of 2 . Let Qo = d, 
QO = D. 
Then suppose that the system of the two spheroids is in uniform rotation about the 
axes of £ with an angular velocity u>. 
The potential ft of the centrifugal forces is given by 
But 
Hence 
n — \oT Qrf + £ 2 ). . . 
z = d, Z = D — £, d + D = c" 
tl = V J '= —V 
f = » X=-( J 
(33) 
(34) 
D = l co 2 (y 2 + z~ — 2zd + d 2 ) 
= i [— i f - r) + i f - k x2 - if) +1 if + r + f — 2zd + d ~\ 
Then, remembering that 
and if we put 
we have 
iv 2 = z 2 — \x 2 - \y 2 , w l = Z, 
q, = x 2 - y\ Q, = X 2 - Y\ 
ft = — \oA h + -l(ohv. 2 — oTdiv l + ^coV + hoAl 2 . 
(35) 
Similarly the rotation potential, when developed with reference to the other 
origin 0, is 
fl= - ± a *Q 2 + iad W 3 - oTD W 1 + laTR 2 + \oTD 2 . . . . (36) 
