1886.] the revolution of Iimacons and Cardioids about their axes. 17 
X=2] FO)K,0) 1,00) JA a, 
x’=2[ FQ) 1,0) K,0n) J,06) dr. 
pydams re 
Also Dy = ee 
aX _dX'_4mrt 
dn dn Ne 
By (26) this becomes 
i FO) J, (08) =F PAD fda a (0): 
0 
Now if ¢ be the angle which ‘ie radius vector drawn from 
the origin to any point on the curve makes with the axis, 
1 =sin& 
_ 38-1 r 
— & 1 . C . 
Therefore (37) becomes 
ce _ doe (3&1) 
ik PQ), Qf) = 
Mae ( ee (eee!) ——— ) 7,(na) F,(d£) dad. 
Let fai tae , (Aa) dx 
=| eer = ari ad, (Aa) da. 
Put A#=@ in (31a), and differentiating with respect to A, 
we obtain 
” On (26% — nr?) J, (A) dé 
MG) ae 
(6? + r*)* 
“6; a(2a?—1) J, (ra) da 
Jo (a* + 1)* 4 
MEG (ad (NaN 
. du= 4 +{ eee 
Sd EG se EG aa aE 
men 14) wise 
WOls Wier: Ir 2 
