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  II. 
  On 
  the 
  Calculation 
  of 
  the 
  Maximum 
  Force 
  between 
  Two 
  

   Coaxial 
  Circular 
  Currents. 
  By 
  H. 
  Nagaoka, 
  Professor 
  

   of 
  Physics, 
  Imperial 
  University, 
  Tokyo 
  *. 
  

  

  THE 
  problem 
  o£ 
  calculating 
  the 
  maximum 
  force 
  between 
  

   two 
  coaxial 
  circular 
  currents 
  originated 
  in 
  the 
  absolute 
  

   measurement 
  of 
  electric 
  current 
  by 
  means 
  of 
  a 
  balance. 
  The 
  

   formula 
  for 
  calculating 
  the 
  force 
  was 
  developed 
  by 
  Lord 
  

   Rayleight 
  in 
  his 
  investigation 
  on 
  the 
  electro-chemical 
  equiva- 
  

   lent 
  of 
  silver. 
  Recently 
  a 
  similar 
  method 
  was 
  used 
  by 
  

   Rosa, 
  Dorsey, 
  and 
  Miller 
  J, 
  in 
  the 
  determination 
  of 
  the 
  

   international 
  ampere. 
  The 
  interesting 
  question 
  as 
  to 
  the 
  

   position 
  of 
  the 
  coils 
  and 
  the 
  maximum 
  force 
  acting 
  

   between 
  them 
  was 
  taken 
  up 
  by 
  F. 
  W. 
  Grover 
  §, 
  who 
  

   expressed 
  the 
  said 
  quantities 
  by 
  means 
  of 
  Jacobi's 
  ^-series. 
  

   In 
  a 
  note 
  on 
  the 
  potential 
  and 
  the 
  lines 
  of 
  force 
  of 
  a 
  circular 
  

   current 
  ||, 
  I 
  have 
  shown 
  how 
  the 
  expansion 
  in 
  ^-series 
  of 
  

   ^-functions 
  converges 
  very 
  rapidly 
  in 
  calculations 
  of 
  like 
  

   nature. 
  The 
  expression 
  for 
  the 
  mutual 
  inductance 
  between 
  

   two 
  coaxial 
  coils 
  and 
  the 
  force 
  between 
  the 
  currents 
  passing 
  

   through 
  them 
  can 
  be 
  conveniently 
  expressed 
  in 
  terms 
  of 
  q. 
  

   Grover 
  extended 
  the 
  expression 
  for 
  the 
  force 
  to 
  terms 
  in- 
  

   volving 
  q 
  1Q 
  and 
  q 
  x 
  s 
  in 
  the 
  power 
  series, 
  thus 
  increasing 
  the 
  

   accuracy 
  of 
  the 
  expression 
  to 
  decimal 
  places 
  scarcely 
  needed 
  

   in 
  practical 
  measurements. 
  From 
  the 
  integral 
  expression 
  

   for 
  the 
  force, 
  we 
  can 
  by 
  differentiation 
  arrive 
  at 
  an 
  expres- 
  

   sion 
  giving 
  the 
  condition 
  of 
  maximum 
  force. 
  This 
  method 
  

   was 
  followed 
  by 
  Grover, 
  who 
  obtained 
  an 
  expression 
  for 
  

   calculating 
  the 
  maximum 
  force 
  that 
  can 
  be 
  applied 
  for 
  given 
  

   coils 
  in 
  finding 
  the 
  distance 
  between 
  them. 
  The 
  expression 
  

   in 
  its 
  final 
  form 
  is 
  sufficiently 
  convergent 
  to 
  be 
  of 
  practical 
  

   value, 
  but 
  the 
  approximation 
  leading 
  to 
  the 
  value 
  of 
  q 
  which 
  

   corresponds 
  to 
  the 
  required 
  maximum 
  seems 
  to 
  offer 
  another 
  

   solution. 
  Obviously 
  the 
  reduction 
  of 
  the 
  integral 
  involves 
  

   the 
  use 
  of 
  elliptic 
  functions, 
  which 
  can 
  be 
  expressed 
  in 
  terms 
  

   of 
  ^-functions 
  ; 
  there 
  is, 
  in 
  addition, 
  a 
  factor 
  containing 
  the 
  

   distance 
  between 
  the 
  coils. 
  This 
  factor 
  gives 
  rise 
  to 
  a 
  very 
  

   convenient 
  formula 
  for 
  finding 
  the 
  required 
  distance 
  when 
  

   once 
  the 
  value 
  of 
  q 
  is 
  known. 
  Thus 
  the 
  first 
  step 
  is 
  essen- 
  

   tially 
  the 
  evaluation 
  of 
  q. 
  According 
  to 
  the 
  method 
  followed 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  Rayleioh, 
  B. 
  A. 
  R. 
  p. 
  445 
  (1882) 
  ; 
  Phil. 
  Trans, 
  clxxv. 
  pp. 
  411-460 
  

   (1884) 
  ; 
  Scientific 
  Papers, 
  ii. 
  p. 
  278. 
  

  

  % 
  Rosa, 
  Dorsey, 
  & 
  Miller, 
  Bull. 
  Bureau 
  Stand, 
  viii. 
  pp. 
  269-393 
  (1911). 
  

  

  § 
  Grover, 
  Bull. 
  Bureau 
  Stand, 
  xii. 
  pp. 
  317-374 
  (1916). 
  

  

  || 
  Nagaoka, 
  Journ. 
  Coll. 
  Sci., 
  Tokyo, 
  xvi. 
  Art. 
  15 
  (1903) 
  ; 
  Phil. 
  Mag. 
  

   vi. 
  p. 
  19 
  (1903) 
  ; 
  Proc. 
  Math. 
  Phys. 
  Soc. 
  vi. 
  p. 
  156 
  (1911). 
  

  

  