542 Dr. A. Russell on the Effective Resistance 



Now noticing that 



j r ker mr "dr = (r/m)kei' mr 



and frkeimrdr = — (r/m) ker' mr, 



(79) 

 (80) 



we find, by substituting the value of i' given by (78) in (76), 

 that 



A bei' mc — B ber' mc-\- G kei' ???c — D ker' ?nc = 0, (81) 



A ber' ?7ic + B bei 7 mc + C ker' mc + D kei' ?nc = 0. (82) 



nC 



By equating also the integral value \ ^irri'^r of the 



current in the return conductor to the value o£ I given 

 by (69), we get 



A bei' mb— B ber' mb + C kei' mb — D ker' mb = — y bei' ma, (83) 



and 

 A ber' mb + B bei' mb + C ker' ??i6 + D kei' mb = — -=- ber' ma. (84) 



The four equations (81) -(84) completely determine the 

 four constants A, B, C, and D. Hence the current density ■ 

 at all points on the outer conductor is found. 



From (73) and (74) we can see at once that 



, ., d r 2(i- Q-, 



(85) 



where i' 6 is the current density on the inner surface of the 

 outer conductor. We also see from (59 J and (61) that 



e = />*'«+ V lo g 



b 31 • 3 f e 2(I-I' x ) 



e aB« +A *^j5 



3#, . {86) 



where i is the current density on the outer surface of the 

 inner conductor. Thus, by addition, 



7 "NT 



e + e' = p» a +pi' 6 + 2/*!log-^. 



(87) 



Hence writing for i a and i\ their values given by (65) 

 and (68) respectively, and also writing for i cos cot and 

 i sin cot their values in terms of I and "dlfdt from (71) 

 and (72), we get, after a little reduction, that 



«+«'== RI + l|? (88) 



