the Conduction of Heat. 387 



replace q by a- da, and introduce the additional factor J„ (lea), 

 subsequently integrating with respect to a between the limits 

 and x> . Comparing the result with that expressed in 

 (20), (23), we see that 



cr cos nO e-z- '1±t 



V(7rt) 



is a common factor which divides out, and that there remains 

 the identity 



e-P 2 W 



2t 



f* ' ada e~^ J n (lea) l/|f ) = J n (kp) e -»*. . (31) 



This agrees with the formula given by Weber, which thus 

 receives an interesting interpretation. 



Reverting to (30), we recognize that it must satisfy the 

 fundamental equation (1), now taking the form 



d 2 v d 2 v 1 dv 1 d 2 v _ do 



d7 2 + a^ + 'pa^p + ~p'W--di'' ' ' ^ l) 



and that when t = Q v must vanish, unless also s=0, p = a. 



If we integrate (30) with respect to z between +oo, 

 setting q — adz, so that crcosnO represents the superficial 

 density o£ the instantaneous source distributed over the 

 cylinder of radius a, we obtain 



era cos n# T /ap\ - £+£! 



v = 



2t 



•® •""«"» ' - ■ ( 33 ) 



which may be regarded as a generalization of (9). And it 

 appears that (33) satisfies (32), in which the term d 2 vjdz 2 

 may now be omitted. 



In V. Kelvin gives the temperature at a distance r from 

 the centre and at time t due to an instantaneous source 

 uniformly distributed over a spherical surface. In deriving 

 the result by integration from (2) it is of course simplest to 

 divide the spherical surface into elementary circles which 

 are symmetrically situated with respect to the line OQ 

 joining the centre of the sphere to the point Q where the 

 effect is required. But if the circles be drawn round 

 another axis OA, a comparison of results will give a definite 

 integral. 



Adapting (12), we write a = csin 6, c being the radius of 

 the sphere, ,v = OQ sin d' = r sin 6', z = rcos0' — ccos0 ) so 

 that 



qc sin 6 g-(c 2 +*' 2 )/4* (cr sin 6 sin 0' \ ^ cose cose' , N 



«-* — sw»*--H — ¥t — ) e - 2t ■ ( 34 > 



