﻿Force 
  between 
  Two 
  Coaxial 
  Circular 
  Currents. 
  15 
  

  

  Expressing 
  the 
  quantity 
  under 
  the 
  parenthesis 
  in 
  terms 
  

   of 
  ft 
  we 
  obtain 
  

  

  + 
  68052q 
  10 
  + 
  337465? 
  12 
  + 
  1513740? 
  14 
  

  

  + 
  6247665£ 
  16 
  + 
  ....} 
  (I.) 
  

  

  The 
  expression 
  in 
  terms 
  of 
  ft 
  is 
  

  

  F 
  = 
  -^L— 
  j(l 
  + 
  12^-192^ 
  + 
  1232^- 
  563% 
  4 
  

   lOv/Aaft 
  (. 
  

  

  + 
  21648ft 
  5 
  -73600ft 
  6 
  + 
  226944ft 
  7 
  - 
  648189ft 
  8 
  + 
  . 
  . 
  . 
  .) 
  

   - 
  12ft 
  loon 
  - 
  . 
  (1 
  - 
  10ft 
  + 
  60ft 
  2 
  - 
  300ft 
  3 
  + 
  1300ft 
  4 
  

  

  - 
  4884ft 
  5 
  + 
  16320ft 
  6 
  - 
  49920ft 
  7 
  + 
  142500ft 
  8 
  + 
  ....)}. 
  

  

  . 
  . 
  . 
  (I'.) 
  

  

  These 
  formulae 
  include 
  many 
  terms 
  extended 
  by 
  Grover. 
  

   The 
  condition 
  required 
  for 
  the 
  maximum 
  force 
  is 
  simply- 
  

   given 
  by 
  

  

  __ 
  =0 
  or 
  ^— 
  T 
  =0. 
  

  

  This 
  evidently 
  leads 
  to 
  the 
  following 
  relation 
  between 
  two 
  

   integrals 
  

  

  cos 
  6 
  d0 
  

  

  3c 
  

  

  Jo 
  (A' 
  + 
  a 
  s 
  

  

  + 
  £ 
  2 
  — 
  2Aacos#) 
  

  

  cos 
  6 
  dd 
  

  

  (A 
  2 
  + 
  a 
  2 
  + 
  2 
  2 
  -2Aa 
  cos 
  0) 
  I 
  

  

  

  This 
  equation 
  was 
  utilized 
  by 
  Grover 
  for 
  finding 
  the 
  

   distance 
  between 
  the 
  coils 
  when 
  the 
  said 
  condition 
  is 
  

   satisfied. 
  The 
  evaluation 
  of 
  the 
  integrals 
  is 
  not 
  an 
  easy 
  

   process, 
  and 
  for 
  finding 
  q 
  from 
  (4'), 
  we 
  have 
  to 
  assume 
  an 
  

   approximate 
  value 
  of 
  z 
  given 
  by 
  Rayleigh's 
  formula, 
  and 
  

   arrive 
  at 
  the 
  final 
  result 
  by 
  successive 
  steps. 
  The 
  solution 
  

   of 
  the 
  problem 
  can, 
  however, 
  be 
  obtained 
  in 
  another 
  form 
  

   by 
  finding 
  the 
  value 
  of 
  q 
  directly 
  from 
  the 
  known 
  value 
  of 
  

   a/A. 
  

  

  