392 Lord Rayleigh : Problems in 



We will, however, write down the integral corresponding to 

 a uniform linear source coincident with the axis of z. If 

 p 2 = x 2 +y 2 , r 2 = z 2 + p 2 , and (p being constant) rdr=zdz. 

 Thus putting in (56) g = q 1 dz, we get 



q, eW f°° e~r^(ip)dr /r . v 



•=-5ri VF^T (5b) 



In considering the effect of periodic sources distributed 

 over a plane xy, we may suppose 



v cc cos Ix . cos my, ..... (59) 

 or again v x J n (&r) . cos ^0, (GO) 



where r 2 =x 2 -\-y 2 . In either case if we write l 2 + m 2 =k 2 , 

 and assume v proportional to e ip \ (1) gives 



d 2 vjdz 2 = (k 2 + ip)v ((51) 



Thus, if 



P-f ip = ~R(cosu i-i sin a), .... (62) 



where A includes the factors (59) or (60). If the value of 

 v be given on the plane z=Q 9 that of A follows at once. 

 If the magnitude of the source be given, A is to be found 

 from the value of dv/dz when z = 0. 



The simplest case is of course that where k=0. If 

 Yew* be the value of v when z = 0, we find 



v=V#pt e-*-WP); (64) 



or when realized 



v*=Ye-*.<f(pP)coi{pi—z^(p/2)}., . . (65) 

 corresponding to 



v = Y cos pt when z = 0. 

 From (64) 



~ (s) = ^^ ' Yeipt = ** ^ ■ - - ' (66) 



if a be the source per unit of area of the plane regarded 

 as operative in a medium indefinitely extended in both 

 directions. Thus in terms of <r, 



v =,^L_ e i(pt-lTr) e -zsl(i v )^ . . . . (67) 

 or in real form 



v= Tjp e ~^ (pl2) cos {pt-fr-z y( P /2)}, (68) 

 corresponding to the uniform source a cos pt. 



