392 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Following the similar process with the remaining terms of (38), and equating to 
zero the coefficient of Wj c , we have from (40'), (41), and (42'), 
2k+1 7 , 34 , 7 (A\*k + 2! , , /a\3i = » k + i\ F" 1 r A 
h + (t) ~u2\ + * (i) L = 
whence 
AVk + 2 ! 
.e — 
c k\2l 
+ 
<A 3 k + i\ U- 
c/ j = 2 &! i! i — 1 
L;. 
(44) 
By symmetry, the condition that the spheroid A may be a level surface is 
Lf - r, 6 ( 
c rl 21 
AA l = m r + i\ 7 i_1 
2 W f r 2 7T7T '• • • 
(45) 
a\^r + k\ P- 
Multiply both sides of (45) by f [ - ) ^ryy ~—- , and perform S on the whole, and 
substitute from (44); and we have 
(A\»k + 21. _ x /a\* 3 
* € U TUT — ¥ w 
a\ 3r = °° r + 2 ! r + 7c! F 
r —1 
2 ! r ! v ! k ! r - l 
/*\9. M 3 /44\ 3i = 00,, = 0 ° r + r + *! 7 i_1 F -1 7 
' 2 W \ T / i = 2 r =2 r! i! r! A:! i — 1 r — 1 ^ ‘ 
(46) 
Introducing the notation (26) for the series involved in (46), we have 
4 — | e ( c ) ji {k + 1) (h + 2) + I (j) (- 
[h, 2, r]j 
/^r \ 3 /y)\ 3 ;= ® rv*-l 
+ «)*© 7 
(47) 
Each value of k gives a similar equation, and there is a similar series of equations 
with small and large letters interchanged. 
Now put 
UV 
4 — To 6 (t) T + 1) (k + 2) X /c 
L-, = To 6 (~) T + 1) (^ + 2 ) 
and (47) becomes 
J 
(48) 
h — 1 + 
a \* a\ 
(7c + 1) (/>' + 2) V-4/ \c 
3 / 3 i = <= 
+ V 2 
3 \2 
[4 2, r] 
(i + 1) (i + 2) 
if 2 (fc + 1) (7c + 2) 
i-i 
ft». r] Tn x < 
(49) 
