1886.] the revolution of Limacons and Cardioids about their axes. 11 
proper solution of (17) will be X =J,,,(Xé). The complete solution 
of (18), expressed in the usual but somewhat awkward notation, is 
Y=A/J,, (Atm) + BY,, (Acm) 
where 1=,/—1; German writers also employ the symbol X,, (Aum) 
in place of Y,,. But since both these functions may be treated 
as real quantities, it appears to me that the introduction of an 
imaginary quantity in the argument creates such needless com- 
plexity as to constitute a fatal ‘objection to the use of this notation; 
and I shall therefore employ the symbols J,,(A7) and K,, (Ay) in 
the place of J,,(Aun) and Y,, (Aen) respectively. The complete 
integral of (18) may now be written 
Y = AI, (\n) + BK,,(2), 
m 
; C (—) ene i T ‘ 5 nes 
where ils => 2 alm + 4) , E61 COS sin odd po0000 (19), 
2”1'(m+4) (° cosrxnd. dd 
K,,= 4 een ee ehee F 
Sa | (20) 
: 2m +A 
(+6) 2 
The integrals (19) and (20) can be easily shown to satisfy (18). 
They may however be obtained otherwise as follows. Writing « 
for Xm, the ordinary expression for J,,(a) is 
mm 
ax 
2 Jarl (m +4) I "508 (z cos g) sin” ddd. 
Whence (19) follows at once by changing @ into ux, rejecting the 
imaginary factor, and multiplying the result by (—)”. 
A different expression. for [,,(z) may be obtained as follows. 
In (19) put An=2, y=2"u,, , then it can be shown that 
du 
d (a) Te 
Now Re == | “e-neost ag 
* This integral has been evaluated by Prof. J. J. Thomson, Quarterly Journal, 
Vol. xvim. p. 377. 
