A treatise on the theory of Bessel functions, by G. N. Watson.

4-8, 4-81]

DIFFERENTIAL EQUATIONS

125

and a normal solution of this equation of which ekz is a factor must be such that
1 a2V
v a2
is independent of b, and, if the solution is to be one-valued, it must be equal
to - m2 where m is an integer. Consequently the function of p which is a
factor of V must be annihilated by
d2 I d i 2
dp2 pdp + k p2
and therefore it must be a multiple of Jm (kp) if it is to be analytic along the
line p = O.
We thus obtain anew the solutions
ekz C. m. Jm (cp).
sin
These solutions have been derived by Hobson* from the solution ekz.Jo(kp) by Clerk
Maxwell's method of differentiating harmonics with respect to axes.
Another solution of Laplace's equation involving Bessel functions has been obtained by
Hobson (ibid. p. 447) from the equation in cylindrical-polar coordinates by regarding 8/az
as a symbolic operator. The solution so obtained is
cos d\
sin b la f (Z)'
wheref(z) is an arbitrary function; but the interpretation of this solution when cm involves
a function of the second kind is open to question. Other solutions involving a Bessel
function of an operator acting on an arbitrary function have been given by Hobson, Proc.
London Math. Soc. xxiv. (1893), pp. 55-67; xxvi. (1895), pp. 492-494.
4'81. Solutions of the equations of wave motions.
We shall now examine the equation of wave motions
a2 a2 a2V aV a1
(1) x+ + ~ + a c
in which t represents the time and c the velocity of propagation of the waves,
from the same aspect.
Whittaker'st solution of this equation is
(2) V = f(x sin u cos v + y sin t sin v + z cos u + ct, u, v) d dv,. -T7r 0
where f denotes an arbitrary function of the three variables involved.
In particular, a solution is
= J f_.eik(xsinucos v+ysinitsinv+zcosit+ct) F (u, v) du dv,
-7r 0
where F denotes an arbitrary function of u and v.
* Proc. London Math. Soc. xxII. (1892), pp. 431-449.
t Math. Ann. LVII. (1902), pp. 342-345. See also Havelock, Proc. London Math. Soc. (2) n.
(1904), pp. 122-137, and Watson, Messenger, xxxvi. (1907), pp. 98-106.