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



The simplest case is that in which the origin lies in the plane 

 of the two straight wires and is midway between them. We then 

 have b = a and 7 = -|7r, so that 



w 2 m- 1 =(-l) m - 1 -i|r 1 , %^i = 0, 



and then, if c< a, 



M = 8ttc 



1 c lie 3 



- - sin cos <f> + 2^- s sin3 e cos s <£ 



1.1.3c° . r n „ . , 



+ ^ — -, — 7. -r sm J (7 cos 06 + . • 

 2 . 4 . a° 



By § 5, these series are convergent provided that the radius of the 

 circle is less than the shortest distance from its centre to either of 

 the infinite wires. 



If we introduce the additional limitation that (/> = 0, so that 

 now a diameter of the circle lies on a line cutting both long wires 

 at right angles, the last expre?sion for M reduces to 



M = 87r cosec {a - vV - c 2 sin 3 0], (c < a) 



as may be seen by expanding the square root (a 2 — c" sin 2 0)K 



APPENDIX [Added U April 1902]. 



The following method, of obtaining the expression 



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



for the mutual potential energy of the systems S and T, seems 

 preferable to that given in §§ 3, 4 of the foregoing paper. 



I will consider first the case in which every part of 8 is further 

 from than any part of T, so that s > t. 



Let the potential due to T be expressed at points on its axis 

 0T (Fig. 3) by the series 



F = £» + g + g+.... (r>t) 



Then the potential due to T at a point H, which lies, at a 

 distance R from the origin, on a radius making an angle yjr witb 

 0T, is 



V R + M* + R 3 + ' V 



where X = cos i/r. 



