538 Dr. A. Eussell on the Effective Resistance 



When mx is very large we shall use the formulas 



mx 



kerm*= \J J=le ^co S g|+f), . . (50) 



. mx 



kei, M =-^^/^sin(^| + |), . . (51) 



mx 



Wm ,.= Vs/MSK)' • • ( 52 > 



7. Formulte containing loth functions. 



We require the following formulse, also, in our solutions. 

 When mc is small, 



S c = ber' mc ker' mc + bei / mc kei 7 ??ic 



711 C 



= — (a + f-logmc) 



, w 6 c 6 .-, v, 37m 6 c 6 - . 



+ 12764 (a ~ l0g mc) + 2 2 . 4 2 . 6». 8 ^ * {d±) 

 and 



T c = bei' mc ker' mc — ber' mc kei' mc 



1 7T m 2 c 2 m 4 c 4 / K t - X 



= -2 + 4-— —48 (55) 



When mc is very great, we may write 



cos mc a/2 /k „, 



& °= to^T' (56) 



and T sin mc \/2 ff , . 



2mc v 7 



8. TA<? complete solution. 



On the assumption that i follows the harmonic law, we see 

 that the complete solution of the equation (A) is 



i = (A ber mr + B bei mr + C ker mr + D kei mr) cos at 



-f ( — A beimr + B ber mr — Ckeimr + D ker mr) sin <w£, (58) 



where A, B, C, and D are constants. 



