572 
MR, G. H. DARWIN - ON - PROBLEMS CONNECTED 
Second. In finding the terms dependent on A ?+1 , dp, <fi +2 it will be better to 
subdivide the process further. 
1 1 d'V'- 
(l) a =—. — (cr— rn-z- 
v 7 7 ua 1 2 I v 7 cfe 
a 2 7(7+1) (27+ 3) 
u 21(27 + 5) 
l o o\dU 
(«-O s 
(50) 
<”•> <"=UlWk 
. A [ b' + 1) (* + 3) „o,- + 3 d / _ _ o; _ 
+1 
(»riA»T W ,°i+3. 2/—1TT\ ± ,^+3_/ r -2/-lTT\ 
“ i/l 1(27+5) ffofi 7 27+5 dod 7 
Then 
since 
(/+ 3) (i +1) — I = i~ + 4 i + 3 — 2 r - 47 - 3 = - i~, 
therefore 
1(274-5) dx V ’ 
• (51) 
This completes the solution for a . 
Collecting results from (49), (50), and (51), we have 
/ I // | /// 
a —a / +a 4-a 
« 3 J(7 + 1) 3 0 c/U 
u |_4(7— 1) dx 
i(i + l)(2i + 3), j o\dU 
21(27 + 5) ^ ~ V 7 dx 
1(27 + 5) dx 
(52) 
Then collecting results, the complete value of a as the solution of the second 
approximation is 
a=a o+ a o'+ a + a ,- 
So that it is only necessary to collect the results of equations (31), (with Y' written 
for Y), (36), (43), and (52), and to substitute for U its value from (39) in order to 
obtain the solution required. The values of /3 and y may then at once be written 
down by symmetry. The expressions are naturally very long, and I shall not write 
them down in the general case. 
The radial velocity p is however an important expression, because it alone is 
necessary to enable us to obtain the second approximation to the form of the spheroid, 
and accordingly I will give it. 
It may be collected from (33), (37), and by forming p and p / from (43) and (52). 
I find then after some rather tedious analysis, which I did in order to verify my 
solution, that as far as concerns the inertia terms alone 
