﻿438 
  Mr. 
  J. 
  Hollingworth 
  on 
  a 
  Physical 
  

  

  If 
  R 
  and 
  Gare 
  not 
  zero 
  terms 
  of 
  the 
  form 
  J 
  ^/(K.ip-\- 
  G) 
  x 
  

   will 
  occur 
  in 
  the 
  solution 
  which 
  are 
  complex, 
  and 
  so 
  give 
  

   rise 
  to 
  a 
  change 
  of 
  phase, 
  and 
  therefore 
  to 
  a 
  loss 
  of 
  power 
  

   in 
  ohmic 
  resistance. 
  

  

  It 
  mny 
  be 
  of 
  interest, 
  however, 
  to 
  see 
  how 
  far 
  the 
  solution 
  

   by 
  elementary 
  methods 
  corresponds 
  with 
  the 
  case 
  in 
  which 
  

   a 
  single 
  conducting 
  plate 
  in 
  an 
  infinite 
  dielectric 
  is 
  considered. 
  

  

  Radiation 
  losses 
  will 
  now 
  of 
  course 
  occur. 
  (The 
  following- 
  

   analysis 
  is 
  chiefly 
  deduced 
  from 
  the 
  results 
  in 
  Macdonald's 
  

   ' 
  Electric 
  Waves,' 
  § 
  57 
  sgg.) 
  

  

  For 
  the 
  general 
  case 
  of 
  an 
  infinite 
  cone 
  with 
  oscillations 
  

   of 
  a 
  form 
  to 
  give 
  rise 
  to 
  circles 
  of 
  magnetic 
  force 
  with 
  their 
  

   centres 
  on 
  the 
  axis 
  of 
  the 
  cone, 
  we 
  have 
  

  

  f= 
  -q 
  2 
  (£fj 
  n+i 
  (K.r,) 
  | 
  J_._j(Kr.) 
  -*<»+*>« 
  J 
  n+i 
  (Kr) 
  } 
  

  

  N 
  sin 
  (n 
  + 
  i)ir 
  d/*i 
  d/* 
  

  

  the 
  sources 
  being 
  given 
  in 
  position 
  by 
  r*=r 
  u 
  6 
  = 
  6 
  1 
  (r 
  x 
  <r), 
  

   summation 
  being 
  for 
  all 
  values 
  of 
  n 
  fo 
  1 
  * 
  which 
  P»(/z, 
  ) 
  — 
  

   (Macdonald, 
  §62). 
  ^ 
  

  

  In 
  the 
  case 
  of 
  the 
  plane 
  O 
  = 
  ~ 
  this 
  reduces 
  to 
  

  

  -2 
  Z 
  g-) 
  J^iCKn) 
  [ 
  J_ 
  w 
  _, 
  (Kr) 
  -^+iKI 
  K+i 
  Kr} 
  

  

  X 
  M 
  . 
  *".. 
  sin 
  ^ 
  1^1^, 
  . 
  . 
  (27) 
  

  

  N 
  Sin 
  (n 
  + 
  i) 
  7T 
  ByLtj 
  dyu. 
  ' 
  v 
  ' 
  

  

  n 
  being 
  any 
  odd 
  integer. 
  

  

  7\ 
  small 
  and 
  1 
  = 
  9O° 
  give 
  case 
  of 
  small 
  circle 
  of 
  sources 
  

   inside 
  small 
  hole 
  in 
  centre 
  of 
  disk. 
  

  

  oo 
  / 
  1 
  \ 
  s 
  T 
  2s 
  

  

  Now 
  J. 
  + 
  |(*)=d-+*25 
  

  

  2^+2^(^ 
  + 
  1 
  + 
  ,)' 
  

  

  i. 
  e. 
  since 
  quantity 
  under 
  summation 
  is 
  finite 
  largest 
  term 
  is 
  

   given 
  by 
  n 
  = 
  l, 
  all 
  others 
  being 
  negligible 
  in 
  comparison. 
  

   Therefore 
  (27) 
  reduces 
  to 
  

  

  since 
  ^— 
  - 
  =1 
  for 
  all 
  values 
  of 
  //, 
  

  

  _—qr 
  1 
  

  

  K* 
  V- 
  

  

  2 
  f 
  -V^ 
  

  

  — 
  {j_ 
  f 
  (Kr) 
  + 
  iJs{Kr) 
  } 
  sin 
  0, 
  (28) 
  

  

  