Prof. Thomson on Peristaltic Induction of Electric Currents. 143 





4(1+/J< I 



£ = — X aC 



DO _ft£(x+2ia--f)2 A:c(.r + 2ia + g)^ 



p(/)(?. = 2 (-l)'{e ^(1+-^^' -e" *('+/J' } 



(11). 



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

 two periodical functions of (^—x) and {^ + x) ; and each of the func- 

 tions JT and (£ is a periodical function of a' simply. The expressions 

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

 expression of periodical functions in trigonometrical series, are: — 



F(/){^. t) 

 !(/)«. 0= 



i(f)(x, t- 



_2 /4(1 



-ay — 



+f)-,it ■=- _'!!l!Mf . ivx . zV? 

 - 2 £ a i^e Sin — sin — ^ 



kc 





"^ 2«-L kc 



2 7 4(1 +/M ^ 

 a \/ kc J 



Tf 



id^kc sin — 



1)*^ . (22-l)7r? 

 -^^^ — sm ^ 



Kl2). 



_ 1 r4(l+/M^-0) 



^ 4a- L 



Ac 



T^l{2i 



!>- 



2a 2a 



;-i)=7r2fi+/)(<-e) 



(2i-l)„ 

 2a 



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 ( 1 2) 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 / increases, the first set of 

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

 vergent enough to be extremely convenient for practical computation. 



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~'"'sin?w the effects of 

 the initial arbitrary electrification of the wires, and employing a 

 method given by Professor Stokes* to express the effects of the va- 

 riations arbitrarily applied at the free ends of the wires. 



Case II. — Three-wire Cable. 

 The equations of mutual influence between the wires may be clearly 

 put under the forms 



fyi=8'i+/(?2 + ?3). cv^=(li+/(q3 + qi)' ci;3=j3-H/(?, + ?2); 



* See Cambridge Phil. Trans, vol. viii. p, 533, " On the Critical Values 

 of the sums of Periodic Series." 



