1886.] the revolution of Limacons and Cardioids about their axes. 15 
By means of the theorem* 
(ee= iA | 226 (2) Hi Ne Ie QENAWEN on. (28), 
it follows that 
ice = [fo ees » (AX) J, (AE) dadn. 
But K,() = ee Rein oe (29) ; 
—e- if ADA Me ee (30) 
Os = Ve Oy) 
and ya-18 : Se eACOLACY oy te (31). 
Secondly, consider the term Aw; the surface condition is 
f F(a) J,(az) dn = — 
Way tM cal ee 
ese 
(Ge le 
SS Nae) 
Aol i “Crepe a0) J (nf) dada. 
In (29) put Aa = 8, then ea) 
* OJ,(0) dé 
MAG? 
K, sae anf OS, OJ,(9) dé addy al ad ,(rAa)dx k 
Ki= 
(7+ (Se x 0 (1 ir eh) p) 
a of O,(0) dé | 
a le GREEN 
aes ee Af ae = Nina ide 
K,= + K, = el GARE AA eee (31 a). 
eI] is 3 s 
eae VE, (A) Jy (a dn eipiejetateisiatar ale lets 32); 
(+1) t f (A) Jy (AE) (32) 
* See Proc. Camb. Phil. Soc., Vol. v. p. 427. 
+ Heine Kugelfwnctionen, Vol. 1. p. 197. 
