536 



Dr. A. Russell on the Effective Resistance 



given by Kelvin. When, however, the core is hollow * or 

 when we wish to take into account the effects of the return 

 conductor, the complete solution has to be found, as the 

 above solution cannot be made to satisfy all the boundary 

 conditions. 



5. The ker and kei functions. 



Imitating Kelvin we shall write the second type of solution 

 of (B), namely K (mr \/t), in the form ker mr -f- 1 kei mr. 

 Substituting x sj i for x in equation (2) and equating the 

 coefficients of the real and imaginary terms on the sides 

 of the equation, we find that 



ker x — [a — log x) ber x 4- (7r/4) bei x 



4 g 



"~ (1 + ^)22 42 + (1+2 +"3 +4)22 42 £2 G2~~'--' (3°) 



and 



kei x = (a— log x) bei x — (7r/4) ber x 



+ 2 2 ( "*~ 2 + 3 ^2 2 .4 2 .6 2 

 where a=0'1159315.... 



f ..., 



(36) 



6. Approximate formula for the ker and kei functions. 



When x is small, we may use the following approximate 

 formulae, which can easily be proved by (5) and (6) : — 



, , it mrx / o i n 



ker mx = a — log mx + j • — r (a 4- J — log m#) 



64 4 , 2 2 .4 2 .6 2 ' 



■ (37) 



kei ma? = --+(« + l-log?n^)-j- + - . — -( a + i^_ 1 



t , 1 . 7T ma? . j_ sm s <2? 3 7r m 5 # 5 



kerwia;= + t-^ — (a + f — logmaA-— - _____ _ 



mx 4 2 v 4 y 16 4 2 2 .4*.6 



and 



kerw# = (a-j-i— Wm^)— - + — . — — 

 to / 2 4 16 



\ fflV /00 . 

 . . . (39) 



_( a + J_l g m ^)____, 



When m# is small we see that 



(40) 



kerw^ = a— log ma? and kei m#= — 7r/4, approximately; 



and hence, when mx is very small, we may write 



her 2 mx -{-kei 2 mx = (log mx) 2 . . . . (41) 



^ * The solution given in Mascart & Joubert, vol. i. p. 719 (1897), for 

 the effective resistance of a hollow inner core is incorrect. 



