1891.] in the Linear Conduction of Heat. 247 



but this interpretation is also liable to objection. For the distri- 

 bution of sources and sinks corresponding to the initial temperature 

 in any single element of the rod is not in itself equivalent to that 

 element. Hence Mr Hobson's solution gives no idea of the part 

 played by the initial temperatures of the separate elements of the 

 rod, and in fact it does not even give a correct result if the effect 

 of the initial temperature in any part of the rod is considered 

 apart from that in the rest. 



The problem is best solved by starting with a single instan- 

 taneous source at one point of the rod, since from this the more 

 general solution can be obtained by integration. 



Consider then the effect of an instantaneous source Q of heat 

 generated at the point x at the time 0. The temperature due to 

 such a source if there were no boundary would be 



Q {x-xj 



exp — 



*Jife * 4ft " 



Let this expression be denoted by v 2 , and let the temperature 

 when the boundary is taken into account be v where 



v = v 1 + v 2 . 



If the external medium be at temperature zero, the boundary 

 condition gives, when x = 0, 



dv 7 n 

 ax 



„ dv., , (d ,\ Q (x-x'f 



Hence -y- 2 — hv = - -^ — h) 9 ,— r exp - 



dx \dx J^Jirkt 4ft 



Q f x' ) x' 2 



= 2jm I m + h \ exp - m when ■ ■ °- 



But v 2 must be the temperature due to a series of images on 

 the negative side of the origin, hence the conditions of the problem 

 will be satisfied by taking 



dv., 7 Q f on + x' , 1 (x + x'f 



^"^ = 27sr"w- +A r xp --4ft- 



_(d ,\ Q (x + xj 



"U + Jvs exp T" 



_fd \ Q (x + x'Y 



'{dx'VYjm^ ^ 



2hQ (x + x'f 



+ 2 v / 7r ft eXP 4ft ' 



