﻿112 
  Mr. 
  J. 
  H. 
  Michell 
  on 
  the 
  

  

  Substituting, 
  we 
  get 
  

  

  /0> 
  *) 
  - 
  —2 
  \ 
  /(ft 
  ?) 
  cos 
  («*-«) 
  cos 
  (n£-e) 
  

  

  */0 
  JO 
  Jo 
  J-oo 
  

  

  x 
  cos 
  m(f 
  — 
  a?)c?f 
  rf£ 
  rf/n 
  d« 
  

  

  Oy2 
  Z' 
  00 
  r»» 
  /»0O 
  

  

  + 
  — 
  1 
  / 
  (ft 
  J) 
  i»V-*»^*+0 
  cos 
  m 
  (?-#)</£ 
  rf? 
  rfiw, 
  

  

  7r 
  y 
  Jo 
  Jo 
  J-* 
  

  

  which 
  is 
  the 
  theorem 
  on 
  which 
  the 
  rest 
  of 
  the 
  present 
  paper 
  

   is 
  built. 
  It 
  is 
  curiously 
  easy 
  to 
  give 
  an 
  a 
  posteriori 
  proof 
  of 
  

   the 
  theorem. 
  Using 
  the 
  value 
  of 
  e 
  given 
  above 
  we 
  find 
  

  

  cos 
  (m— 
  e) 
  cos 
  (wf— 
  e) 
  = 
  cos 
  nz 
  cos 
  n% 
  — 
  cos 
  n(z 
  + 
  ?) 
  —. 
  — 
  ^— 
  s 
  , 
  

   Integrate 
  the 
  last 
  two 
  terms 
  with 
  respect 
  to 
  n, 
  viz. 
  

  

  »-'0 
  

  

  and 
  the 
  quadruple 
  integral 
  becomes 
  

   2 
  

  

  7T 
  2 
  

  

  rfX> 
  /*oo 
  /»oo 
  /^»oo 
  

  

  I 
  \ 
  1 
  1 
  / 
  (ft 
  C0S 
  UZ 
  C0S 
  n 
  f 
  C0S 
  m 
  (f 
  "" 
  #) 
  °? 
  ^? 
  <^ 
  ^ 
  

  

  Jo 
  Jo 
  Jo 
  J 
  — 
  oo 
  

  

  -- 
  /(ft$)w^- 
  Am2(z+f) 
  cosm(ft~a?)^^^m. 
  

  

  w 
  Jo 
  Jo 
  J-«j 
  

  

  The 
  former 
  integral 
  is 
  /(#, 
  2), 
  and 
  the 
  latter 
  disappears 
  

   with 
  the 
  triple 
  integral 
  in 
  the 
  given 
  formula. 
  

  

  Considering 
  now, 
  for 
  simplicity, 
  the 
  water 
  infinitely 
  deep, 
  

   it 
  appears 
  at 
  once 
  that 
  the 
  required 
  solution 
  for 
  <f> 
  is 
  

  

  , 
  2v 
  C°° 
  f°° 
  f 
  00 
  f 
  00 
  ... 
  cos 
  (ns 
  — 
  e) 
  cos 
  (nf— 
  e) 
  /f 
  . 
  x 
  , 
  ., 
  , 
  2 
  

  

  * 
  = 
  ^ 
  /(ft 
  8) 
  — 
  i/4-r-ii 
  — 
  ^cosm(f-^>-^-+^ 
  

  

  77 
  Jo 
  Jo 
  .0 
  J-00 
  vrrir 
  + 
  r? 
  

  

  X 
  d%d%dmdn 
  

  

  2^3 
  f«> 
  ^co 
  /too 
  ffte-nfrfltz+Q/g 
  . 
  

  

  ~ 
  ^J 
  J. 
  J_/ 
  (f 
  ' 
  r) 
  7^^i 
  Sm 
  -t-0 
  + 
  -^-W-j, 
  } 
  

   + 
  ^Jo 
  X 
  J-, 
  /(fe 
  g) 
  •i-»v/ 
  J 
  « 
  cos 
  -ft—)*^"** 
  #**»; 
  

  

  (5) 
  

  

  