384 Lord Rayleigh : Problems in 



If in (12) we put q — cr da and integrate with respect to a 

 from to go , we obtain a solution which must coincide with 

 (7) when in the latter we substitute z for x. Thus 



f adae-«V*n (^) = 2texV±t f . . (13) 

 a particular case of one of Weber's integrals*. 



It may be worth while to consider briefly the problem of 

 initial instantaneous sources distributed over the plane (f =0) 

 in a more general manner. In rectangular coordinates the 

 typical distribution is such that the rate per unit of area is 



a cos 1% . cos mrj (14) 



If we assume that at ,r, y, z and time t v is proportional to 

 cos Ix .cos my, the general differential equation (1) gives 



dv . m i 2\ c ^° 



or 



TtV v J = d?V V J' 



so that, as for conduction in one dimension, 



e -z 2 /4t 

 v = A cos Ix cos my e~( l2 + m2 )t — ~r — ? • • (15) 



and 



v dz = 2^/ir . A cos Ix cos my e~ (^ 2 + w2 K 



J 4 



Putting £ = 0, and comparing with (14), we see that 



a 



V* 



(16) 



By means of (2) the solution at time £ may be built up 

 from (14). In this wa} r , by aid of the well-known integral 



i 





coszcx dx= 6 , . . (17) 



a 



we may obtain (15) independently. 



The process is of more interest in its application to 



* Gray and Mathews' Bessel's Functions, p. 78, equation (160). Put 

 72*=0 ; A=0. See also (31) below. 



