1886.] the revolution of Limacons and Cardioids about their axes. 13 
11. The properties of the J functions are so fully discussed in 
Todhunter’s Functions of Laplace that it is unnecessary to consider 
them here, further than to note that the principal equations which 
they satisfy are the following, viz. 
pn Git 
Jn i ao Ji, a, Baa 
5) 
J, =I. aN Buereve\avetslelo os abaseneverstevers (24), 
m-1 a ™ 
le a J, ys) 
where x is the argument and the accents denote differentiation. 
By means of the definite integrals (19) and (20), it can easily 
be shown that the J and K functions both satisfy the equations 
5 
yb = = Ihe a Wha | 
a 
fe yh a8 | hy Sos acta Glau a (25) 
wv 
A =—T, J 
12. Again, since J,, and K,, both satisfy the equation 
/ 2 
ax a 
it follows that 
ee Pr Mla ate = Ck mL, JL) = 0 3 
C(—)” 
x ’ 
Te ig Fe 
m 
where C'is independent of . Substituting for [,’, KX,’ their values 
from the first and second of, equations (25) respectively, we 
obtain 
mt1 
U iped C8 Ww ISG. S = Ae) ’ 
and ME oan LUE a = ) 
a C 
whence LE,- LK,=- a 
C 
therefore ik — 1K, = = 
