EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
391 
The term in L- t in the second of (38) gives us, as in § 2, 
4 JAy 31^-' <* Wi 
377(1 \ c) 2i — 2 A 3i + 1 ' 
(42) 
The potential due to rotation is ^cohv 2 or -g-w 2 W 2 , being the second term of (35) or 
(36); this term we find it convenient to write 
icoW 
(43) 
The sums of the several terms (39), (40), (41), (42), and (43) are to be regarded as 
the potential of perturbing forces by which the spheroid a or the spheroid A is dis¬ 
turbed, and the arbitrary constants l and L are to be so chosen that each may be a 
figure of equilibrium, 
We may consider the spheroid a by itself, and the solution for it will afford us the 
solution for the spheroid A by symmetry. In order to find the disturbance, the formulas 
(40) and (42) must be transferred. 
Now by (4), with i = 2, 
4t tA 3 1 /Ay c 3 W % _ 4t tA 3 r _ /Ay * = » & + 2! /aV w k 
Sc T<L \7) X ~~ 3c >'[e) ;., 0 2U! \c) <?' 
(40') 
And again, by (4), 
4t -a? (AySLiT 1 - 1 4t rA 3 a /a\s* = » Jc + i\ p- 1 UCA w k 
3c \c) 2i — 2 PA + l 3c 2 \c) * = „ AXk\ i^l Li \c) a? 
(42') 
Then (39), (40'), the sum of (41) from k = co to k = 2, the sum of (42') from i = oo 
to i = 2, and (43) together constitute the disturbing potential, all now referred to the 
origin o. 
In order to find the disturbance of the spheroid a, we add the perturbing potential 
to f7 ra 3 /r, give r its value (38) in this term, put r = a in the perturbing potential, and 
make the whole potential constant by equating to zero the coefficient of each 
harmonic term. 
We will begin by putting rja = 1 + wq/r 3 , and considering only the perturbing 
potentials (39) and (43). We have then, for the coefficient of w 2 /r 3 , 
4_„2 l,i l,& n 2 i 1,,2/vS 
— 3 ua • 3 e ~T 4 W a "T 6 W 00 • 
Now, with the value of e in (37), 
— fvm 3 . = — iV® 8 and — -/ 2 - -f i -f i = 0. 
Hence the coefficient of ttq/r 2 vanishes, and the term in e in (38) has been properly 
chosen to satisfy the perturbing potentials (39) and (43). 
