288 PROFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
a. = 
1 _ 
1 - 2p L 
1 
, 4(1-pf 1.3...27.-3 ■ 
— p 02H-2 + ^2,1-2 + 2.4... 2?. - 2 ^ ~ 
(^2n — 
1 - 2A 
(1 - X)' K 
2,x- 
2n\^ 
2 n-l 
^ 2 , 1-2 
By successive applications, starting from the values for 7^ = 0, I find 
Ar 
1.3 
1 , fig — p, A^ — 4 p (1 -p — 2 4 P (^ + P + tp")’ 
" p ( 1 + P + fp^ + fp^); ^10 — 2 8 P P ^P^ ^P" 3 WP'^): 
2.4.6 
1 ...9 
^12 — 9 ''' 10 ^ + p + fp® + fp'^ + -^p"^ + "isp^)^ 
a. = 
— 
1, Og _ 1 -}- X, (^4 _ 1 -j- X fi- fX^, _ 1 -f- X + fX~ + fX®, 
«s = 1 + X + yX'^ + f X^ fi- 
1 + X + f X2 + f X3 + + ^x^ 
Br. = 
= 
1 + ..pfj + ^2 — P + ^4 — ip 
1 + P + ~2 (^ ~ ^P) 
B, 
B 
1 .3 
2.4^ 
1.3.5 
8 
1 + P + Ip^ + 3 ^^.o (1 + P — P^) 
iP 
10 
Bi2 — 
2.4.6 
1 ...7 
2” . .18^ 
2^... 9 
^ 7.10 ‘ 
12 
1 + P + Ip® + fp® + - o (1 + p + ip® — P®) 
00 )“ 
16 
1 + p + fp® 4- Ip® + Ap^ + (1 +p + Ip® “ ip® — ip^) 
20 
1 + P + Ip® + 9 P® + 2 ~TP‘^ + ifsP^ + 9^2 ( ^ + P + 7 P® + ip® — ilp"^ — 7 P^) 
?>n = 
hr. = 
a = 
a = 
a 
O' — 
\^n - 
1 - lx, 63 = 1 + X — X^ = 1 + X + |X= — X 3 , 
1 + X + fx* - ix* - fXi, 63 = 1 + X + fx^ + ixi* --nxi - 4 X». 
0 + -+—+ 
^ ^ 3o;2 ^ 30)2.7/2 ^ 
ip 1 + P + 3^,^,2 (1 — ip) + ~4j 
2.4 
1 4 P + fp'^ + 9^2 4 P ~ p^) + ^ (1 — ip + 4p~) 
9,fi 
