﻿10 
  Dr. 
  J. 
  W. 
  Nicholson 
  on 
  the 
  Relation 
  of 
  

  

  where 
  

  

  and 
  

  

  C 
  = 
  - 
  3-^ 
  r 
  Z"^) 
  C 
  io(cos 
  lo' 
  + 
  e-">yiw 
  

  

  =--'r©''(l)('-I)-I 
  

  

  so 
  that, 
  when 
  a- 
  is 
  positive, 
  

  

  I 
  dio 
  . 
  { 
  sin 
  {w^ 
  — 
  Gio) 
  + 
  e-^«3_cr«- 
  j. 
  

  

  with 
  the 
  associated 
  equation, 
  when 
  p 
  is 
  positive, 
  

   j 
  dw' 
  {sin 
  (w^ 
  + 
  pw) 
  + 
  e-'^^+P'" 
  j- 
  

  

  (13) 
  

  

  Writing 
  

  

  Vi 
  = 
  ^^' 
  

  

  3 
  

  

  so 
  that 
  cr=3X^, 
  and 
  combining 
  (10) 
  and 
  (12), 
  we 
  obtain 
  the 
  

   curious 
  result 
  

  

  Jx(2A,3) 
  =i!- 
  C 
  J^o{3^cos(w;3~3X2^(^)-sln(w;3_3;^2^•)_^-^.3-3A2«;}^ 
  

  

  or 
  finally 
  

  

  J,(2X^)=J^ 
  r"c?i(;^2cos(z(;3-3\\(; 
  + 
  i7r)-e-'-^^^^"}. 
  . 
  (14; 
  

  

  In 
  a 
  similar 
  manner, 
  

   J_,(2\^) 
  =^\ 
  div\2cos(io^-^3Xhc-i7r) 
  + 
  e~^''-''^''^-\, 
  . 
  (15) 
  

  

  Ii(2XS) 
  =^- 
  i 
  "6Zic]2cos(i^H3\2w. 
  + 
  ^7r)-^— 
  H3A^-}, 
  . 
  (16) 
  

  

  1_^^(2\3) 
  = 
  - 
  J^ 
  \ 
  dw{ 
  2 
  cos 
  {w^ 
  4- 
  BX^i^; 
  - 
  J 
  tt) 
  + 
  ^--^+?'A'^^ 
  \^ 
  (17) 
  

  

  