Mr Searle, On the Coefficient of Mutual Induction, etc. 405 



Now let there be at H a single particle, m, of the system S. 

 Then the potential energy of T in the presence of m is mV or 



m mP,(X) mP 2 (X) /rt . 



ffo £+gi — ar-^+ft & +••• • (* >«) 



r=t 



Fig. 3. 



But the potential due to to at a point K on 02 7 , for which r< R, 

 is mjHK, or, when /fiT is expanded in ascending powers of r by 

 the fundamental expansion formula of zonal harmonics, 



L 



1 ,-. p i( x >, + p *&> \ ' 



(R>r) 



If there be any number of particles in S, then, provided they 

 be all further from than any part of T, the potential energy 

 of T in the presence of S is given by 



^ m ^ mP 1 (X) ^ mP 2 (X) . . 



W = ff %^ + g a £ J + g 2 X j± J + ..., (s>f) 



while the potential due to S, at a point on OT near the origin, is 



the summation in each case including all the values of R and X 

 necessary to take account of every particle of S. 



Comparing the last two expressions, we see that if the potential 

 due to 8, at points on OT near 0, be denoted by 



U=Y + rY 1 + r"-Y 2 + ..., (r<s) 



then the mutual potential energy of S and T is 

 W = g Y +g 1 Y 1 +g 2 Y 2 + ..., 

 the series for W being absolutely convergent provided s >t. 



29—2 



