Self-induction of Wires. 4o5 



subject to (178). Here p = — ??i 2 /RS, so that the state of the 

 line at time t after it was V , when left to itself, is got' by 

 multiplying each term in the expansion by e~ m2f / RS . The 

 corresponding current is given by RC=— dV/dz. But the 

 solution thus got will usually only be correct, although (180) 

 is correct, when there is, initially, no energy in the terminal 

 apparatus. If there be, additional terms in the numerator 

 of (180) are required, to be found by the energy-difference 

 method of Part III. They will not alter the value of the 

 right member of (180) at all ; they only come into effect after 

 the subsidence has commenced. Similar remarks apply what- 

 ever be the nature of the line. It is, however, easy to arrange 

 matters so that the energy in the terminal apparatus shall 

 produce no effect in the line. For example, join the two 

 conductors at one end of the line through two equal coils in 

 parallel ; if the currents in these coils be equal and similarly 

 directed in the circuit they form by themselves, they will not, 

 in subsiding, affect the line at all. 



Returning to (177), or other equivalent expression, it is to 

 be observed that particular attention must be paid to the 

 roots ml = 0, which may occur, or to the series of roots p 

 belonging to the m = case, when we are working down from 

 the general to the special, and happen to bring in m = 0. 

 Take Z x =0 for instance, making, by (175) and (160 b), 



0=-Z o -~ ianml, 



where m 2 =—BjoR". Then 



dp dp 2m \ dp p J 2 \ dp p ) ' 



Now, as long as Z is finite,, m cannot vanish ; but when Z 

 is zero, giving ml= any integral multiple of 7r, m = is one 

 case. Then we have, when m is finite, 



d4> i/dR"W\ A d<f>' i d, .p,,. , 1fto . 



^ = 2U^ + 7> and ^ = 2^ R > ; - ' (182) 



but when m is zero the middle term on the right of the pre- 

 ceding equation becomes finite, making 



dcf>/dp = l(d'R"/dp). 



The result is that the current solution contains a term, or 

 infinite series, apparently following a different law to the rest, 

 with no corresponding terms in the V solution. This merely 

 means that the mean current subsides without causing any 



