302 
PROFESSOR G. II. DARWIN ON THE PEAR-SHAPED FIGURE 
Also 
Then since 
we find 
1 1 
qo 4 + /3cr 2(5] + 1) 
, with g-Q = 1 . 
P^iy) = = I5v{v^ — 1 ), 
(!)• 
V o _ 1+ff 
(^) + Ayg'PsS (z/)), with g/ = 
The equation for /3cr is 
iyS 2 { 3 , 2 }{ 3 , 3 } 
or 
^cr + 4/3.3.4 = 
;8cr -|- 6/8 = 
8_| + 
If we write 
‘1+ \^a' 
(&)= = 1-1/3 (1 - /3), 
the proper solution of the quadratic equation is 
/3cr = 4 (P. — 1 — |]8). 
Also 
'A 
1 
— Q , — W z' Al V ^ = 3 -^^, with g/ = 1. 
8 + /So- 4(74 +1 — 1/3) i/f jP 
Then since P^ {v) = f (5w — 1) — 1)4 P^ (^) = 15 (w — l) (w — l)% "^ve find 
■ ■ -(4 
P 3 I {v) = I (P, - 1 + 2/3) (w - ^ -^5 
1 -/S 
5 ( 1 -^) 
6 - = 1 , type OOS and {v) = (v) {v), with q^ 
The equation for is 
^cr — 1/3.3.4 = 
f/3^{3, 2} {3, 3} 
8 + jSa 
A comparison with the last case shows that we have only to change the sign of /3 ; 
accordingly if 
(^^ 3 )^ = 1 + f /8 (1 + / 8 ), 
we have /So- = 4 (Pg — 1 -j- f^S), 
4(5, + 1 + f/S)’ 
with gi ■— 1 . 
