314 PROFESSOE G. II. DARWIN ON THE PEAR-SHAPED FIGURE 
The following integrals may also be verified by differentiation : 
rsm«t _ r(4 - ^ (3 - 4.r/) _ (3 - 2r/) - 
J AfiA 2riY~ {k^ - (f) 2/cY'2'“ 2/dr/2'-(«' - 2 b ^ 
A sin-vir cos W , . 
0 • 
+ 
^0/0/0 0\\; 
2'2" - 2") -^1 
C sin® , 
- if) - f(2 -J>f) „ , if - 1 s. , 1 + 2f-KH2r/^ 1) 
-LJ- 1" n.4.,/± "l" ir\ rs ! , , r^ -tj 
22b'‘ - 2b 
'2rj^c/‘^ 
+ 
- 2b 
A tan t/t a sin ^Jr cos -v/r 
/.■> / 4. 
K'-q^ 
22'M/c=-2bAi 
o • (27), 
siid 
A7Af 
/cMl-22b-2Hd-52b 
22b'^(^^ - 2b^ 
22Y“ - «bl - 22b 
.» /.■> I r> /■) 
22*2 * "t 
-:aAA-2AA JT I 
^ 2/cbY~(«'-2b 
sin -v/r cos xjy 
n n /.-) o rn / n Ov 
2/<-a: ~2"2 * (^' — 2") 
A sin -yjr cos t/t 
/c'2 (/c2 - 2-FA 22'^ (- 2b-Ai- 
(28). 
r sin® W , 2 — 7-1 2(1 — K-K~) „ sin W cos-vl/- A tan-v^ /on\ 
I f \3 ^ ^ ”” 4 '4 ^ '4 \ ”* '4 .\'"^/ 
J COS" /C^/C " K^/C ^ K-fC /C ^ 
J cos- -v/f A® 
Now in (26) we have to put 
K- = q- 
4 — 5q~ 
in (27) 
and in (28) 
- _ ^3 2 - 52- _ 
^ 1 - 42^ ’ 
,3 4 - 
9n2 
1-22^ 
Introducing these values, and taking the integrals between the limits y and 0, we 
find : 
1 3,A 2 - 2 fi^M 72 '~ - 1 -' 1 ^ + 5 ?'- - 1 r , (Y~ - 0 A sin 7 cos 7 ] 
.... (30). 
sin" 7 [_ LK-Cfq 
-<rl 
P,>Q 3 ' and P/Q/ = ^ ^ 
Sin" 7 [ Iq^q 
1 — 72 * — (1 — 52“ — 22 ^) sin- 7 \ A tan 7 
2)dw^ ; ■ 
1^3 
T ^ and 33 3a 3 - _ (l -22b(2-32 b 2 - llgY ! ^ 
As and 4^3 as — ,n,e1 m .. 2 /c^vi 
(31). 
sin" 7 L bK-q^q'^ ' QK~K '~q*q 
n — 5q- + 62 ^ - 2 “ (2 — ll 2 b'b shi- 7 \ sin 7 cos 7 ] 
fow ) AAd I • 
