1886.] the revolution of Limacons and Cardioids about their axes. 9 
Let e(v—1) =n’, 
2n (n+ 1)=’e, 
and let c and m increase indefinitely, whilst v approaches in- 
definitely near to unity, but so that both the quantities 7 and » 
remain finite. 
Equation (11) becomes on transformation, 
(1 +7) (e+ a et) _ an (n+ Do =) 
dn’ dy C 
whence proceeding to the limit we obtain 
CO Oe aie 
dag * dq = 
which is a form of Bessel’s equation. 
‘ ie dd 
Again, Q,=| aie 
a @, 0 (v+/v*—1 cosh d)"™ 
e [ exp {— (n +1) log (v +. /v*— 1 cosh $)} dg. 
Now 
(n +1) log {v + /v* — 1 cosh ¢} 
2 2 
=4{14/20%¢ + 1} log {1 aay e 4 cosh sh 
=n cosh 
ultimately ; 
On | e~mrcosh bh = ¥, (Nem) seseeseseaeeee- (12) 
0 
ultimately; and is therefore a Bessel’s function of the second kind. 
In the same way we can show that 
eo) == |" eros So 
=e CNL) hgh Bseccn ee. cor sh cdentis (13), 
and that Po (w= = | " gud cos tdd w<l 
‘ 0 
SF (sd) SAR odo (14) 
ultimately. 
