280 PROFESSOR O. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
Thus 
sm-/3, 0 I ox 
(r + 
= cos^/3 — Tj + sin'/3sin- 6 — (cos^/3 — rj cos-<p + (cos~/3 — k ~ — /c-r^) siu"(9cos'^. 
Let us assume, as we know to be justifiable, 
{y~ + z^) = AS, + -b C 
= - - C] + [A./o^ + Bq'A] K= siu 2 e 
+ + B^j”] COS' (^ — [A + B] K~i< ~ siu' 9 cos^ <^.. 
If we equate the coefficients of sin” 9 and cos” </> in these two expressions, we have 
Aq'^ + Br// = 
sin* yd 
AqJ' + BqA = 
Tj — cos^ /3 
The solution of these equations may be written 
A = —- h - 
2D2,^ \ D + 2D2',^ ’ 
-n _ _ _ 1 _ A I cos” /3 \ 
“ y ^ I) - K-)'^ 2Dc/q 
The simplest way of finding C is to jDut sin^ ^ —A cos^ (f) = ^, so that 
/C"* ^ 
S, ~ S,^ = 0 ; we thus find 
A = x(l- + cos”/3) — It,. 
Now for brevity write 
ij- 
k' 
L = 
sin yd 
1 — 
cos” /3 
cos yd cos 7 \ 1) T- K- 
AVe then have 
M = 
sin /3 
4D(/o” cos yd cos 7 \ — /c- 
1 + 
cos- yd '' 
A = iL, b = -2 
sin d o“ 
Hence, substituting for its value. 
cos d cos 7 
sin d 
7^2 
Now 
sin® d 
3 sin2 d 
C0s2 d C0s2 7/1 do 1 do2 ^ 
2 . .'2 \ 21^ 2 I 3 
17 
i 7 siiF d \ 5 V A^Fd Ai'^Fb ' ' 
- P = i iU f-Y (cp _ 2 tT) 
ilTcWdcf) ^ \kj ^ ^ 
Therefore 
