54 PROBLEMS IN THE CONDUCTION OF HEAT If in (12) we put q^o-da and integrate with respect to a from 0 t we obtain a solution which must coincide with (7) when in the luttt substitute z for x. Thus o a particular case of one of Weber's integrals*. It may be worth while to consider briefly the problem of initin stantaneous sources distributed over the plane (Ģ=0) in a more ge manner. In rectangular coordinates the typical distribution is such tha rate per unit of area is a- cos It; . cos mij ............................... (] If Ave assume that at x, y, z and time t, v is proportional to cos lx . eo the general differential equation (1) gives _ __, * or ~ so that, as for conduction in one dimension, 0-Z2/.K v = A cos lx cos my e-P+m*>t j , .................. (* Y t r+cc and vdz = 2^/7r.Acoslxcos my e J 00 Putting t=Q, and comparing with (14), we see that cr 2V7T' By means of (2) the solution at time t may be built up from (14) this way, by aid of the well-known integral e~a*x- cos 2crc <Ģe = e-flS/Ŧ2 t I -x a ' ....................^ we may obtain (15) independently. The process is of more interest in its application to polar coordii If we suppose that v is proportional to cos nQ . Jn (kr), d?v I dv 1 d-v also*(?l)aybelonwd ******' ^^ ^^ P' ^ *^ (16°^ Put "=°' ^d integrate with respect to z from oo to + oo , we may recover (9).(9) are of the standard form if we take