the Conduction of Heat. 385 



polar coordinates. If we suppose that v is proportional to 

 cos n6 . Jn(kr), 



d 2 v Idv , 1 d 2 v 72 /1D . 



d? + 7-dr + 7>dF- = - kv > • • • < 18) 

 so that (1) gives 



dv 72 d 2 u, 



. . (19) 



and 





v = A cos nO J n (hr) e—kH — y— # . . 



■ (20) 



From (20) 





p%i*=2t/ir.Acosn0J,(fr)«-**. . . (21) 



«y — oo 



If the initial distribution on the plane £ = be per unit 

 area 



<rcosn0JnQcr), (22) 



it follows from (21) that as before 



A -*£ ^ 



We next proceed to investigate the effect of an instan- 

 taneous source distributed over the circle for which 



?=0, %=acos(f), r) = as'm <fi, 



the rate per unit length of arc being q cos ncj). From (2) 

 at the point x, y, z 



Jo 8tt 3 / 2 ^ ' * • ' W 



in which 



ri=(£-x) 2 +{ v -ijy + z' = a* + P 2 + z*-2a P cos{cl>-6), 



if x=p cos 6, y = psin0. The integral that we have to consider 

 may be written 



I cos ncj) ep' cos (<P- 9 ) dcf)= cos n(d + yjr) ep' co *^d^ 



= cos nO I cos nifr e? cos $ dty— -sin nO I sin m/r ep' cos '^ d\jr, 



.... (25) 

 where i/r = <£ — #, and p' = ap/2t. In view of the periodic 



