394 
PROFESSOR G. H. DARWIN ON FIGURES OF 
The potential of the inequality — y 0 -e B' 2 wjr 2 in the first of (53) is 
47ra 3 g x S 2 %’ 4 
5 • a r 5 — 
1 o 3 M 3 S 2 w 4 
• (54) 
The potential of the similar inequality in the second of (53) is 
4t rA 3 /AV x eWW i 
3c 
1 o' 
W 
The term in mj L in the first of (53) gives us 
47rA 3 3m* (a\ k fa\ l + 1 + 2 
3c 2k — 2 \ cj \r 
The term in M t in the second of (53) gives us 
4t tA s /a)* 3 Jlfj d 8 2 W i+2 
3c \c) 2i — 2 
22 s 
2i+l 
(55) 
• • 
• (56) 
• (57) 
Lastly, the sectorial term itself is 
! o o 
“ ; W 
(58) 
The sums of the several terms (54), (55), (56), (57), and (58) are to be regarded as 
the potential of perturbing forces by which the spheroid a, or the spheroid A, is 
disturbed, and the arbitrary constants m, M, are to be so chosen that they may each 
be figures of equilibrium. We may consider the spheroid a by itself, and the solution 
for it will afford the solution for the spheroid A by symmetry. In order to find the 
disturbance, the formulae (55) and (57) must be transferred. For this purpose we 
require the second transference formulae. 
By (16), with i— 2, we have for (55) 
4t rA 3 /Ay 1 c s SAF, 4t rA 3 -, fAy k = °° 7 + 2! fa\* $ho t+s , ,, 
3c \c) 10 ~ e -S 5 3c l6e \c) t .? 2 0!7 + 2!\c/ a k ’ (°° ’ 
And by (16) we have for (57) 
4t tA s fay 3Mj c ! 8 3 JF i+s 
4vrA 3 , /a\ 3i ^°° k + i\ r i_1 ir fa\ k 8ho k+ „ , , x 
= - “3V U4 < h ^2!i+2! ^ fi) — • (° 7 ) 
Then (54), (55'), the sum of (56) from Jc = co to Jc = 2, the sum of (57') from 
i = co to i ==■ 2, and (58) together constitute the disturbing potential, all now referred 
to the origin o. 
