the Conduction of Heat. 395 



The series (84), unlike (77), does not terminate. It is 

 ultimately divergent, but may be employed for computation 

 when z is moderately great. 



In these periodic solutions the sources distributed over 

 the plane, sphere, or cylinder are supposed to have been in 

 operation for so long a time that any antecedent distribu- 

 tion o£ temperature throughout the medium is without 

 influence. By Fourier's theorem this procedure may be 

 generalized. Whatever be the character of the sources with 

 respect to time, it may be resolved into simple periodic 

 terms ; and if the character he known through the whole of 

 past time, the solution so obtained is unambiguous. The 

 same conclusion follows if, instead of the magnitude of the 

 sources, the temperature at the surfaces in question be knovvn 

 through past time. 



An important particular case is when the character of the 

 function is such that the superficial value, having been con- 

 stant (zero) for an infinite time, is suddenly raised to another 

 value, say unity, and so maintained. The Fourier expression 

 for such a function is 



1 1 r°° sin pt 1 



iK\ 



(85 



the definite integral being independent of the arithmetical 

 value of t, but changing sign when t passes through 0; or, on 

 the understanding that only the real part is to be retained, 



1 1 r e** , 



We may apply this at once to the case of the plane c = 

 which has been at temperature from t—— co to t = J and 

 at temperature 1 from t = to t = oo . By (64) 



1 f 00 e ipt-z«(ip) 



dp. . . . (87) 



■ = i + 



P 



By the methods of complex integration this solution may 

 be transformed into Fourier's, viz. 



r—jwr^ m 



2 r /2V< , 



\ZlT 



%J0 



which are, however, more readily obtained otherwise. 



