412 Mr Wilson, On Convection of Heat. 



putting x — ut for x in the expression last obtained. So that 

 we have 



{x-utf+y* 



s P 6 = 



^irk't 



wt 



If then a continuous line source of strength Q per unit length 

 per second starts at t = — oo and lasts till t = the rise of tem- 

 perature due to it when t = will be given by 



, f) ( ce-utf+y 2 



^=1 ArS 4k ' 1 dt. 



kirk't 



Consequently putting x" + y 1 = r' 2 , 



snd = -^i e 2 *' — - dt. 



To solve the problem then it is merely necessary to evaluate 

 the integral 



IvH r 2 \ 



t dt, 



r 



which we shall denote by y. 



Let x = tt, and a = -rz-, , 



4&' 4fc 



f e'^L u dy f°rM- 



so that y = I — i dt, hence ~ = — I — - — a£. 



Now let at = - , so that dt = dz, 



z x 



d 



an 



Hence V^ = — - I — cfo — -^ I e ^ z dz, 



dx 2 x J # 



This equation may be written 



d?y _a 1 dy 



d 2 y dy „ 



or x -=-°- + — — ay = 0. 



dx- dx 



