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



§ 4. To find the potential energy of Q in the field of S, 

 I follow a method suggested by § 131 b in Maxwell's chapter on 

 Spherical Harmonics. We have simply to multiply the mass of 

 each particle of Q by the potential at that particle due to S. 

 That is, if U denote the potential due to S, we must find the sum 

 XmU. 



Now, whatever be the form of the system 8, we know that at 

 any point for which r < s, where s is the least distance of any 

 particle of S from the origin, the potential can be expanded in the 

 "absolutely" convergent series of spherical harmonics 



U=Y + rY 1 + r , iY 2 + v ..(r<s) (5), 



when Y , Y 1 ... are functions of the two angular coordinates 0, <f> 

 employed to fix the direction of the radius vector r. 



Hence when the angular coordinates of the axis OT are 6, <f>, 

 the potential due to $ at a point on OT at a distance h from 0, is 



U^Y + hY 1 + h'Y. 2 + ..., 

 so that 



W = imU=2,m(Y Q + hY 1 +h*Y 2 +;..y 



= Y Xm + F, Xmh + Y. 2 XmJi 2 + .... 



Hence by (4) 



W = g Y + g 1 Y 1 + g. 2 Y. 2 + (6). 



By what has been proved in § 3, this series expresses the mutual 

 potential energy of the systems 8 and T. 



We see that any term of the series, as g n Y n , is obtained by 

 multiplying g n by the value of Y n corresponding to the direction 

 of the axis of T. 



§ 5. It is easily shewn that, under the condition s > t, imposed 

 by § 3, the series (6) found for W is absolutely convergent. For 

 since s > t, we can take a length q such that s> q >t, and then 

 each of the series 



Y + qY 1 + q*Y 2 +..., 



9o/q + 9i/q 2 + 9o/q s +..., 



is absolutely convergent. Hence, using | | to denote the numerical 

 magnitude of a quantity, 



| Y n+1 1 Y n | < 1 /q, | g n+1 lg n \<q, 

 so that | Y n+1 g n+1 /( Y n g n ) | < 1 . 



The last result shews that the series for W is absolutely con- 

 vergent if the least distance from of any particle of S exceed 

 the greatest distance from of any particle of T. 



