248 Mr 0. H. Bryan, On Conduction of Heat. [June 1, 1891. 



Therefore by integration 



Q (x + x'f . r 2hQ „ (1 4- a/)" 7f . 



In the last integral put g — oc + z. Also add fl 1} thus the whole 

 temperature assumes the form 



Q f (x-aTf (x + x'f) 



v = v.+v= ^— y=^exp - , T . + exp- V 



-2/i( 



Qe~ fe (m + z + aff , 

 - — . exp — . ,- — - . dz. 



The first line represents the temperature due to the given source 

 and an equal source at the geometrical image, the second line 

 represents the temperature due to a line of sinks extending from 

 the geometrical image to infinity in the negative direction. None 

 of the images of the source lie on the positive side of the origin. 



For the temperature due to the initial distribution v = <$>(x) we 

 write (p(x')dx' for Q, and deduce by integration, 



1 r\ {x-xj (x + xj) ,,,.,, 



2h r 00 / -00 (x + x'f 



" zjm J. I, exp (_ fe) • exp — m * w **• 



This solution holds good whether <f> (x) be finite and continuous 

 or not. 



[Note by Mr Hobson. — Mr Bryan's formula is undoubtedly an 

 improvement on the one I gave in the paper referred to. It should 

 be observed that his formula may be obtained from mine by 

 integration by parts.] 



