406 Mr Searle, Oh the Coefficient of Mutual Induction, etc. 



If every part of S be nearer to the origin than any part of T, 

 and if the potential due to S, at distant points on OT, be denoted 



r_fc + g+g+.:. 



and the potential due to T, at points on OT near the origin, by 



U =G + G 1 r + G,r 2 +..., 



then we can shew, in a similar manner, that the mutual potential 

 energy of S and T is 



W=G y + G 1 y 1 +G 2 y 2 + .... 



Prof. T. J. I' A. Bromwich has kindly pointed out to me that 

 the connexion between the equations (1) and (2) requires a word 

 of comment. The point is that if S n be the sum of the series (2) 

 to n terms, the quantities 



Lt cos ^ = i (Lt H=00 o n ) and Lt. rt=00 (Lt cos ^ = io 7l ) 



are not necessarily equal unless the series (2) be uniformly conver- 

 gent. It is however easily shewn that this condition is satisfied 

 provided r be definitely greater than t. 



If M be any particle of T, whose polar coordinates with 

 respect to OT are R, <f>, then the potential at distant points on OT 

 is expressed by a series, whose ?ith term is XMR n P n (cos <f>)/r n+1 

 or g n /r n+1 , the summation including all the particles of T. Hence 

 \g n \ "Jf> t n 1 \M\. When T is a system of electric currents, flowing 

 in circles whose common axis is OT, it can be shewn that 



\g n \$47rt n +*%\i\, 



where t is the greatest distance from to any wire of the system. 

 If now we take two lengths k, I, such that k > t and I > k, then 

 | g n P n (cos ty)/l n+1 1 < | g n /k n+1 \ for all values of cos yjr, the extreme 

 values + 1 included. But, by the inequalities just mentioned, the 

 series g /k + g 1 /k 2 + ... is absolutely convergent, and hence when 

 r <£ I the series (2) is uniformly convergent for all values of cos-^r. 

 Thus, provided r be definitely greater than t, we may put cost/t=1 

 either before or after the summation. 



