STABILITY OF THE PEAR-SHAPED FIGURE OF EQUILIBRIUM. 
Hence 
2B =A + 
/ 9 9 1 9 9 
gv g* x--q_x 
Now the coefficient of F (y) in this same combination of integrals is 
2 q x 
1 1 ‘ 
—2 -Ta 2F> Z 
S[x V X 
In this we may substitute for B x its value, and thus find 
f y sin 4 Q 
Jo A/A 
{x 2 -q 2 ) 
F(y)- 
2 g.Y.V-?. 1 ) 
E(y) + 
sin y cos y cos /3 
2q' x 2 {^-q x 2 ) A/' 
This expression together with (l) give all the required integrals, and it only 
remains to tabulate p 1 , A, for the several types of function. 
For the sake of brevity I write 
C x = {q 2 -q 2 ) 2 (qx-qif ■■■{q 2 -q 2 ) 2 (x = l, 2... n), 
the factor which would vanish being in each case omitted. 
When there is only one q , C\ is to be interpreted as unity. 
Table of Values of f, g, h, and A x . 
Type. 
% 
order of 
harmonic. 
s 
rank of 
harmonic. 
f. 
0- 
h. 
A x . 
EEC 
2 n 
2 1 
co 
CO 
CO 
c x 
EES 
2n + 2 
21 
- UUI (K?- q gf 
1 
x'-UqA 
1 
CO 
f lx (k 2 - q x 2 ) C x 
ooc 
2n+l 
2t+\ 
co 
n 
nr/v 
1 
co 
-q'x?C x 
oos 
2 n + 1 
2t+\ 
1 
GO 
co 
OEC 
2 n +1 
2/ 
. 
GO 
GO 
n 
i 
q* 2 C x 
OES 
2 n + 3 
2 1 
- K*K‘m (k2 - q *)2 
1 
k'hVv 
1 
1 
q.v 2 q'x 2 (x 2 -q x 2 )C x 
EOC 
2n + 2 
2t+l 
CO 
n 
n 2 y 
1 
- iV 
1 
- q x 2 q'x 2 C x 
EOS 
2n + 2 
2t+\ 
n 
K-n (/c2-^2)2 
1 
CO 
- 
1 
-q x 2 (x 2 -q x 2 )C x 
c 2 
