1 •*--"- H PKOBLEMS IN" THE CONDUCTION OF HEAT 57
"we integrate (30) with respect to z between ± oo , setting q — ordz, so cosn# represents the superficial density of the instantaneous source over the cylinder of radius a, we obtain
/oox
(38)
may be regarded as a generalization of (9). And it appears that satisfies (32), in which the term d2v/dz* may now be omitted.
"V. Kelvin gives the temperature at a distance r from the centre a,-fc time t due to an instantaneous source uniformly distributed over B.pHerical surface. In deriving the result by integration from (2) it is of simplest to divide the spherical surface into elementary circles which ftx.-© symmetrically situated with respect to the line OQ joining the centre of trie s;pliere 0 to the point Q where the effect is required. But if the circles bo drgfwn round another axis OA, a comparison of results will give a definite
-A.cietpting (12), we write a = csm0, c being the radius of the sphere, = O Q sin &' = r sin 6', Z — T cos 6' — c cos 6, so that
/i M2L»»v;.* •/!•/!/ fO COS 6 COS 6'
qc sin 0 Q- (*+*->& /cr sm ^ sin B\ ----- « -
^- — —
Tliis has now to be integrated with respect to 6 from 0 to TT. Since the reen.lt> must be independent of 6', we see by putting & — 0 that
70 (p sin (9 sin 0') eP™*6™36' sin 0 dd o
the simplified form and putting q = crcd0, where a- is the superficial don. si-fey, we obtain for the complete sphere
(c-r? (c+r)«
v~ r~i~ \e ~~e I'
-* 4 \ /
with (6) when we remember that Q = 47rcV.
will now consider the problem of an instantaneous source arbitrarily dista'ilb'u.ted over the surface of the sphere whose radius is c. It suffices, of course, to treat the case of a spherical harmonic distribution; and we suiTrpose that per unit of area of the spherical surface the rate is Sn- Assuming bhsa/fc v is everywhere proportional to 8n, we know that v satisfies
d { . ndv\ . 1 d*v . , . ,^ rt
.{o()
^—3 -s-Ta ,
sin 0 d0 \ d0J sin2 0quare plate may be treated precisely upon the lines of tl paper referred to. The potential energy of bending per unit area has tl expression