428 Mr. 0. Heaviside on the 



the summation being with respect to the p's which are the 

 roots of <f>{p) = } without inquiring too curiously into its 

 strict applicability, or bothering about equal roots. Here p 

 has to be d/dt and the p's the roots of 



= %o — fi r i) = 0; 

 so that (162) expands to 



c ^ s (X+ g Y)(X 2+g Y 8 ) 



A |^~^ It~P 



(164) 



where the single q takes the place of the previous q or q l7 

 which have now equal values, and C has the same ex- 

 pression on both sides of the seat of impressed force. But 

 e 2 is constant with respect to t, whilst C is initially zero ; 



Lence t, = *fl-*0 



d/dt —p —p ; 



which brings (164) to 



c=s (X +y Y )( X +? Y^ 2(1 _ 6p() (165) 



which is the complete solution. By integration with respect 

 to z we find the effect due to a steady arbitrary distribution 

 of e put on at £=0 ; thus 



w 1 eivdz 

 0=S-*^(l-O, • • • • (166) 



where (t> f = d(f)[dp, and w> is the normal current-function 

 X + </Y. To express the V solution, turn the first w into u. 

 The extension to e variable with t, as in Part III., is obvious. 

 But as the only practical case of e variable with t is the case 

 of periodic e, whose solution can be got immediately from the 

 equations (162) by putting p 2 =— n 2 , constant, the extension 

 is useless. Note that q and q 1 are not equal in (162), and 

 therefore in the periodic solution obtained from (162) direct 

 they must be both used. 



The quantity — <f> which occurs here is identical with the 

 former complete 2(U — T) of the line and terminal apparatus 

 of (157) or (158). 



