the Conduction of Heat, 383 



In Kelvin's summary linear sources are passed over. If: 

 an instantaneous source be uniformly distributed along the 

 axis of: z, so that the rate per unit length is q, we obtain at 

 once by integration from (2) 



q dz e-(* 2 +^+y2)/ 4 ; q e -^2 +!/ 2 )/4t 



8^/2^3/2 - 4irt " -' W 



»/ — CG 



From this we may deduce the effect of an instantaneous 

 source uniformly distributed over a circular cylinder whose 

 axis is parallel to g, the superficial density being <x. Con- 

 sidering the cross-section through Q — the point where v is 

 to be estimated, let be the centre and a the radius of the 

 circle. Then if P be a point on the circle, OP = a, OQ = ?-, 

 PQ = p, ZPOQ = #; and 



p 2 = a 2 + r 2 — 2ar cos #, 

 so that 



-r 



aa _ 



±7Tt 2t 



e 



-'.©■ • m 



I (V), equal to J (ix), being the function usually so denoted. 

 From (9) we fall back on (8) if we put a = 0, 2iraa = q. It 

 holds good whether r be greater or less than a. 

 When x is very great and positive, 



I »w=7(L-r < 10 > 



so that for very small values of t (9) assumes the form 



era - £n£0? 



V ~2 s /{irrat) e " ' 



vanishing when t = 0, unless r = a. 



Again, suppose that the instantaneous source is uniformly 

 distributed over the circle f=0, f=acos<£, 7) = a sin <j>, the 

 rate per unit of arc being q, and that v is required at the 

 point a?, 0, z. There is evidently no loss of generality in 

 supposing 7/ = 0. We obtain at once from (2) 



2 " qad$e-^t . 



87r 3/2£3/2 > .... (11) 



where 



r 2 = (%- x) 2 + v 2 + z 2 = a 2 + x 2 + z 2 -2ax cos $. 

 Thus 



from which if we write q—adz, and integrate with respect 

 to z from — cc to -f co , we may recover (9). 



.-( 



