1911] PROBLEMS IN THE CONDUCTION OF HEAT 55
so that (1) gives
dv d"v '
a-Z*t
and w = A cos nd Jn (kr) e~m —j- ...................... (20)
From (20)
ro vt& = 2V^.4cos»0J~«(&r)e-*3t ................ (21)
—CO
If the initial distribution on the plane z = 0 be per unit area
acosn6Jn(Jcr), ........................... (22)
it follows from (21) that as before
We next proceed to investigate the effect of an instantaneous source distributed over the circle for which
£ = 0, £ = c& cos </>, 77 = a sin $, the rate per unit length of arc being q cos n<p. From (2) at the point x, y, z
I" 27r q cos 9i<6 G~V^ add> . „ . N
— I, 8%»> ' ........................ <2*>
in which
r2 = (| - «)2 + (17 - 2/)s + 22 = a2 + /a2 + ^2 - 2ftp cos (</> - ^),
if /» = p cos 6, y — p sin 0. The integral that we have to consider may be written
I cos n<p ep'cos ^'~6) dd> = I cos n (0 4- -v/r) e/5'208 'o .'
r , r
= cos nv cos n^r efcos>'' dty — sin nd
J .'
where ty = <f>— 0, and p' — apf2t. In view of the periodic character of the integrand, the limits may be taken as — TT arid + TT. Accordingly
T rv
cos nty e^'003^ d^lr = 2 I cos n^lr ep <
—n J(t
/•+TT
sinw-^eP008*^ =0;
•/ — 7T
rZrr /-TT
and / coBTufteP'vHto-QdQ = 2 cos «^ I cos^n/rep'008^^ ....... (26)
. •/ o Jo
The integral on the right of (26) is equivalent to 7rln (p), where
(27)ted the method to be so successful as Ritz made it in the cas of higher modes of vibration.