﻿418 
  Dr. 
  J. 
  W. 
  Nicholson 
  on 
  Inductance 
  and 
  

  

  ^nd, 
  M 
  and 
  N 
  being 
  real, 
  

  

  ^2/M 
  , 
  ^^N_ 
  4a^(log<^/g--'/^Jo/^^JoO(l-/^Ji/^<3^J"/) 
  . 
  

  

  The 
  Bessel 
  functions 
  have 
  an 
  argument 
  ha=xi^^ 
  and 
  if 
  V 
  

   l)e 
  the 
  velocity 
  of 
  propagation 
  of 
  electromagnetic 
  disturbances 
  

   in 
  the 
  outer 
  medium, 
  

  

  h:=nalY 
  (4) 
  

  

  M 
  may 
  be 
  described 
  as 
  the 
  correction 
  for 
  closeness 
  of 
  the 
  

   -wires. 
  

  

  The 
  effective 
  resistance 
  per 
  unit 
  length 
  is 
  given 
  by 
  

  

  Tj_4nw 
  ber.'??bei' 
  .2?-- 
  bei<^'ber' 
  .« 
  ^ 
  ,^. 
  

  

  ^~ir' 
  (ber'^')'+(bei'^)2 
  * 
  * 
  ^^ 
  

  

  Now 
  the 
  functions 
  ber 
  ^, 
  bei 
  x 
  are 
  usually 
  defined 
  by 
  

  

  jQ(a;i^) 
  = 
  ber 
  x 
  + 
  i 
  bei 
  .^% 
  

   so 
  that 
  

  

  Ji(,rti) 
  = 
  — 
  fc-t(ber' 
  X 
  + 
  L 
  bei' 
  x), 
  

  

  Ji(£ct^) 
  = 
  — 
  i(ber^' 
  x 
  + 
  t 
  heV^ 
  of), 
  

  

  ^^^ 
  [M 
  Ji 
  _fju 
  ber\'?; 
  + 
  ^bei^^ 
  

  

  (7) 
  

  

  ka 
  ' 
  J/ 
  X 
  ' 
  ber^'a.' 
  + 
  tbei'^.^'' 
  

   -whence 
  on 
  reduction 
  

  

  (l-/^Ji/y^aJ/)/(l-l-AtJiA^«JiO= 
  (E 
  + 
  iF)/D, 
  . 
  (6) 
  

   -where 
  

  

  E^2 
  = 
  (^ 
  ber^^ 
  xf 
  + 
  (x 
  bei" 
  x)^ 
  - 
  (fi 
  ber' 
  xf 
  - 
  ((jl 
  bei' 
  xY,-^ 
  

   ¥x 
  =2/Lt(ber'^bei".a;— 
  bei'.'??ber".aj), 
  I 
  

  

  Dx^ 
  = 
  [x 
  hev'' 
  xf 
  4- 
  [x 
  bei" 
  .t') 
  ^ 
  + 
  (//, 
  ber' 
  .r)^ 
  + 
  (/^ 
  bei' 
  xf 
  j 
  

   + 
  2/^.3? 
  (ber^ 
  x 
  ber^^ 
  ^7 
  4- 
  bei' 
  .27 
  bei^' 
  x). 
  J 
  

  

  But 
  if 
  accents 
  denote 
  differentiations 
  with 
  respect 
  to 
  x, 
  

   t/= 
  ber 
  ^4- 
  1 
  ber 
  a? 
  

   is 
  a 
  solution 
  of 
  

  

  «o 
  that 
  

  

  X 
  ber" 
  X 
  + 
  1^' 
  bei" 
  x 
  + 
  ber' 
  a? 
  + 
  1 
  bei^ 
  .i' 
  = 
  t^(ber 
  .x? 
  + 
  l 
  bei 
  a?), 
  

  

  and 
  therefore 
  

  

  a?ber''^= 
  — 
  ber'^— 
  ^bei^, 
  1 
  

  

  A* 
  bei'' 
  ti; 
  = 
  .2; 
  ber 
  x 
  — 
  bei' 
  ^r, 
  J 
  

  

  