130 



Each of the functions F and E is clearly the difference between 

 two periodical functions of (%x) and ( +z) ; and each of the func- 

 tions Jf and <JE is a periodical function of a; simply. The expressions 

 for these four functions, obtained by the ordinary formulae for the 

 expression of periodical functions in trigonometrical series, are as 

 follows : 



2 



Either (11) or (12) may be used to obtain explicit expressions for 

 the solutions (10) and (10)', in convergent series ; but of the series 

 so obtained, (11) converge very rapidly and (12) very slowly when t 

 is small ; and, on the contrary, (11) very slowly and (12) very rapidly 

 when t is large. It is satisfactory, that, as f increases, the first set 

 of series (11) do not cease to be, before the second set (12) become, 

 convergent enough to be extremely convenient for practical compu- 

 tation. 



The solutions obtained by using (12), in (10) and (10)', are the 

 same as would have been found by applying Fourier's ordinary pro- 

 cess to derive from the elementary integral e- mt sinnx the effects of 

 the initial arbitrary electrification of the wires, and employing a 



(12). 



