Self-induction of Wires. 429 



Let C be the finally reached steady current. By (166) it is 



*-*(-&)&"*• ••*••■■ (167) 



To this apply (163), with p = 0. Then a finite expression 

 for O is 



Co=Oo/<M \ 'ew dz, .... (168) 

 Jo 



where tv and <f> are what w and <£ become when p = in them. 

 Or, rather, it would be so if q and q 1 taken as identical could 

 be consistent with p = 0. But this is not generally true, so 

 that (168) is wrong. To suit our present purpose, we must 

 write, by (162), 



°o = S Z^{ ( X + ?i Y ) §y* + qoY)dz+(X + g Y)§\(X + qi Y)dz} 



=2,(—p(j)')- 1 < w 1 \ ew Q dz + w \ ew\dz\ ; . . . (169) 



the g being used in w , the q x in w x . Now we can take 

 j? = 0, and get the correct formula to replace (168), viz. 



C =t-< w 10 \ ew QO dz + w 00 \ ew 10 dz\; . (170) 



the second meaning that p = in w and w^ 



If there is no leakage (K = in S") , C becomes a constant, 

 given by 



C =fW*-{ f R^ + R + Ri},. . (171) 



where the numerator is the total impressed force, and the 

 denominator the total steady-flow resistance ; R, R , and R x 

 being what R", — Z , and Z 1 become whenp = in them. 



But when there is leakage (170) must be used ; it would 

 require a very special distribution of impressed force to make 

 C the same everywhere. To find the corresponding distri- 

 bution of Y, say V , in the steady state, we have then 



-dCJdz=KY , 



so that a single differentiation applied to (170) finds V . 

 Knowing thus C finitely, we may write (166) thus, 



