58 PROBLEMS IN THE CONDUCTION OF HEAT
e, co being the usual spherical polar coordinates. Hence from (1) v function of r and t satisfies
dv _ d?v %dv_ n(n +l_)jy = Q
dt ~~ dr- + r dr r2
When ?i = 03 this reduces to the same form as applies in one ditnei For general values of n the required solution appears to be most easily f indirectly.
Let us suppose that Sn reduces to Legendre's function Pn<», \ p = cos 6, and let us calculate directly from (2) the value of v at ti and at a point Q distant r from the centre of the sphere along the axis The exponential term is
if p = refit. Now (Theory of Sound, § 334) J4"' Pn 0*) e<* ^ = 2i» y
whence f +1 PB00 a*^ = 2i*H / (£] J^(- ip), ............. (.
.' —l V \"P/
or, as it may also be written by (27),
Substituting in (2)
we now get for the value of v at time t, and at the point for which
It may be verified by trial that (44) is a solution of (38). Wl is not restricted to the value unity, the only change required in (44) : introduction of the factor Pn (/u).
When ?z = 0, Pn(^) = l, and we fall back upon the case of w\ distribution. We have
Using this in (44), we obtain a result in accordance with (6) in wh representing the integrated magnitude of the source, is equal to present reckoning.6™36' sin 0 dd o