Mr Searle, On the Coefficient of Mutual Induction, etc. 399 



orclinates for the two systems S and T. 

 straight line OT is called the 

 " axis" of the system T; the "axis" 

 thus extends in one direction only 

 from 0. Now let the potential 

 due to T be expressed at any point 

 K on its axis by the series 



V = g -° + ^+ g -l + ...(r>t) 



Then the semi-infinite 



Fig. l. 



(1), 



where OK = r. Then by Legendre's theorem the potential at any 

 point R on a radius OR, which makes an angle -^ with the axis 

 OT, is 



y = 9o P o (COS^ ) | 9iPi (COS jr) + > 



(2), 



the series (1) and (2) being "absolutely" convergent if r>t 

 where t is the greatest distance from of any particle of the 

 system T. 



But (2) is the potential at (r, yjr) due to a system, Q, of singular 

 points of orders 0, 1, 2, ..., and moments g , g 1 , g 2 ... placed at 0, 

 every particle of each singular point lying on the axis OT. Thus, 

 at all points outside the sphere r = t, the potentials due to T and 

 to Q are equal. 



Now, if W be the potential energy of T in the field of 8, 

 the potential energy of 8 in the field of T is also W. But, 

 when s > t, so that every particle of S lies outside the sphere 

 r = t, centred at the origin 0, the potential energy of 8 in the 

 field of T is equal to the potential energy of $ in the field of 

 Q. This last is equal to the potential energy of Q in the field 

 of 8. 



Considering the system Q, let m, mf, ... be the masses of the 

 particles which form the whole system of singular points, and let 

 h, h! , ... be their distances from in the direction OT. Then the 

 potential due to Q, at any distant point on OT, is 



Tr _, m _., /l h h? 



V = % , = Xm - + - +- + 



r — II \r r 2 r A 



Xm Sm/i %mh 2 



.(3). 



But the values of V given by (1) and (3) must be equal. Hence, 

 equating coefficients of the powers of 1/r in the two series, we 

 obtain 



2m = g , Xmh = g x , Sm/i 2 = g 2 , &c. 



(4). 



