﻿1875.] 
  Theory 
  of 
  Elliptic 
  Functions. 
  167 
  

  

  even 
  and 
  \£ 
  uneven) 
  give 
  rise 
  to 
  identities 
  of 
  the 
  same 
  class 
  as 
  (1), 
  and 
  

   which, 
  it 
  appears, 
  are 
  all 
  elliptic-function 
  relations. 
  It 
  is 
  alsop 
  ointed 
  

   out 
  how 
  (1) 
  and 
  the 
  other 
  similar 
  identities 
  discussed 
  in 
  the 
  memoir 
  

   admit 
  of 
  being 
  simply 
  established 
  by 
  ordinary 
  algebra 
  and 
  trigonometry 
  

   in 
  two 
  different 
  ways. 
  

  

  Other 
  identities 
  and 
  formulae 
  are 
  noticed 
  and 
  compared 
  ; 
  ex. 
  gr., 
  for 
  the 
  

   cosine-amplitude, 
  writing 
  

  

  X== 
  2K> 
  z== 
  2K" 
  ( 
  l 
  =e 
  K 
  =e-M-, 
  r 
  = 
  e 
  &=<?-' 
  

  

  we 
  have 
  

  

  cosam 
  u. 
  

  

  I 
  zr~— 
  cos 
  x 
  + 
  ^ 
  cos 
  3x+ 
  % 
  2 
  cos 
  5x 
  + 
  &c. 
  \ 
  

   [1 
  + 
  q 
  1 
  + 
  2 
  I 
  + 
  2 
  5 
  J 
  

  

  \ 
  1— 
  2^cos2^+2 
  2 
  1— 
  % 
  3 
  cos2^+2 
  6 
  J 
  

   2;— 
  sech 
  (2— 
  v)— 
  sech 
  (2+y)-|- 
  &c.} 
  

  

  2kK' 
  

  

  7r 
  f 
  . 
  1 
  _ 
  _ 
  a 
  1. 
  . 
  / 
  cosh 
  v 
  

  

  — 
  — 
  i 
  sech 
  2 
  — 
  4 
  cosh 
  

   2&K' 
  l 
  \cosh2z+cos2*/ 
  

  

  cosh 
  2v 
  

  

  - 
  + 
  &c. 
  

  

  )} 
  

  

  cosh 
  4«+ 
  cosh 
  4*/ 
  

  

  _ 
  7r 
  fsinh(gv— 
  2) 
  smh 
  3(|v— 
  2) 
  ^ 
  sinh 
  5(|j>— 
  2) 
  _ 
  ^ 
  V 
  

  

  \ 
  cosh 
  ^ 
  v 
  coshfi/ 
  coshfi/ 
  ' 
  J 
  

  

  =WK- 
  { 
  SeAZ 
  - 
  TFr 
  C 
  ° 
  Sh 
  Z+ 
  if? 
  eosh3 
  Z 
  -&e. 
  } 
  ; 
  

   so 
  that 
  if 
  x, 
  2, 
  v 
  be 
  any 
  four 
  quantities 
  subject 
  to 
  the 
  relations 
  

  

  2 
  7TX 
  

   — 
  , 
  2 
  = 
  

  

  P 
  

  

  he 
  identities 
  are 
  

  

  ^so 
  that 
  07= 
  

  

  VOL. 
  XXIII. 
  

  

  sech 
  x 
  — 
  sech 
  (x 
  — 
  — 
  s 
  ech 
  + 
  ju) 
  + 
  sech 
  — 
  2^t) 
  

   + 
  sech 
  (#+2yu) 
  — 
  <fcc. 
  

  

  = 
  sech*-4cosh*f 
  ^g^U, 
  uT^w 
  + 
  *>) 
  

  

  Vcosh 
  2# 
  + 
  cosh 
  2/x 
  cosh 
  4# 
  + 
  cosh 
  4/x 
  y 
  

  

  _ 
  2 
  I 
  sinh 
  (J/i 
  — 
  a?) 
  _ 
  sinh 
  3(j/x 
  — 
  a?) 
  

  

  1 
  cosh 
  cosh 
  J-yu 
  

  

  _^sinh5(|ju 
  — 
  a?) 
  _ 
  ^ 
  c 
  1 
  

  

  cosh-||Li 
  J 
  * 
  * 
  ^ 
  w 
  

  

  1 
  4 
  cosh 
  # 
  . 
  4 
  cosh 
  3x 
  „ 
  

  

  = 
  seen 
  a? 
  + 
  — 
  — 
  &c. 
  

  

  e^+1 
  e 
  3 
  ^ 
  + 
  l 
  

  

  _2x 
  j 
  cosh 
  2 
  _j_ 
  cosh 
  32 
  cosh 
  5* 
  ^ 
  1 
  

  

  \coshj»/ 
  coshf*/ 
  cosh 
  I 
  v 
  ' 
  J 
  

  

  2?r 
  f 
  sinh 
  if 
  sinhfv 
  . 
  « 
  1 
  

  

  = 
  _ 
  COS 
  2 
  j 
  -— 
  . 
  2 
  - 
  f 
  8 
  ■ 
  + 
  &C. 
  V 
  

  

  a 
  L 
  sm 
  2 
  # 
  + 
  sinh 
  h 
  v 
  sin 
  a? 
  -f 
  sinh 
  %v 
  J 
  

  

  