1911]' PROBLEMS IN THE CONDUCTION OF HEAT
When n = 1, Pa (/u.) = /it, and
59
,(*,)= / —
A •' *' W«
sm a? a?
- cos #4; ...................(47)
and whatever integral value n may assume /,l+j is expressible in finite terms.
We have supposed that the rate of distribution is represented by a Legendre's function P7l (//,). In the more general case it is evident that we have merely to multiply the right-hand member of (44) by Sn, instead of PB.
So far we have been considering instantaneous sources. As in II., the effect of constant sources may be deduced by integration, although the result is often more readily obtained otherwise. A comparison will, however, give the value of a definite integral. Let us apply this process to (33) representing the effect of a cylindrical source.
The required solution, being independent of t, is obtained at once from (1). We have inside the cylinder
v = Apn cos nd, and outside v — Bp~n cos nd,
with Aan = Ba~n. The intensity of the source is represented by the difference in the values of dv/dp just inside and just outside the cylindrical surface. Thus
a-' cos nd = n cos n& (Barn~l + Aan~r),
whence Aan = Ba~n = a-'a/Zn,
a' cos nd being the constant time rate. Accordingly, within the cylinder
a- a fp
A //10.
fl = — - - cosn0, ........................... (48)
I2n \aj ^ '
and without the cylinder
_^a fP\~n V~2n (a)
cosnO............................(49)
These values are applicable when n is any positive integer. When n is zero, there is no permanent distribution of temperature possible.
These solutions should coincide with the value obtained from (33) by putting cr = a-' dt and integrating with respect to t from 0 to oo . Or
C?i5
~^ 1
n Va/
.(50)
the + sign in the ambiguity being taken when p < a, and the — sign when p > a. I ha,ve not confirmed (50) independently. ndv\ . 1 d*v . , . ,^ rt