414 Dr. T. J. I'a. Bromwich : Examples of 



§ 2. Some Applications of Heaviside's Formula. 



A simple case whicli can be dealt with at once by the aid 

 of the results of § 1 is the following : — 



(8) A sphere is heated to uniform temperature u — u , and 

 allowed to cool, the surrounding atmosphere being at 

 zero. 



Here u — u is initially zero ; and, proceeding as before, 



we find that 



r , N sinh [qr/c) ,_,._ 



-(u -u) = . I* ' J u , .... (28) 

 c y suing v J 



taking for brevity the case in whicli the surface-conductivity h 

 is treated as infinite. 



Then the mean temperature is given by 



Mo -fi = 3l-cothq-- 2 \u Q (29) 



To interpret (29), we observe that the operator is the 

 same as in (8) ; and accordingly we need only substitute 

 in (4) from equations (9), (10), and (11). The result is 

 easily found to be 



u - 



fl 6 ^ 1 -nWcflt/c*\ 



u = woi i — 2 ^ ~V r 



I tt 2 n= i n 2 J 



It will be noticed that, as might be anticipated, the 

 series (30) can be derived from (12) by differentiation with 

 respect to t. 



Further examples of Heaviside's formula can be derived 

 similarly from the other problems of § 1. 



(e) Problems of induction-balances. 



Electrical problems, in which the arms of a balance 

 contain inductances and condensers (in addition to resist- 

 ances) have been considered repeatedly by Heaviside * ; 

 it is hardly too much to say that the formula (4) was con- 

 structed with a view to such problems — and Heaviside's 

 own discussion of the formula is based upon electrical 

 considerations. 



* For example, see ' Electrical Papers/ vol. i. p. 412, vol. ii. pp. 256, 

 280, etc., and many other papers. 



