﻿166 
  

  

  Mr. 
  J. 
  W. 
  L. 
  Glaisher 
  on 
  the 
  

  

  [Jan. 
  14, 
  

  

  January 
  14, 
  1875. 
  

  

  JOSEPH 
  DALTON 
  HOOKER, 
  C.B., 
  President, 
  in 
  the 
  Chair. 
  

  

  The 
  Presents 
  received 
  were 
  laid 
  on 
  the 
  table, 
  and 
  thanks 
  ordered 
  fcr 
  

   them. 
  

  

  The 
  following 
  Papers 
  were 
  read 
  : 
  — 
  

  

  I. 
  " 
  On 
  a 
  Class 
  of 
  Identical 
  Relations 
  in 
  the 
  Theory 
  of 
  Elliptic 
  

   Functions/' 
  By 
  J. 
  W. 
  L. 
  Glaisher, 
  M. 
  A., 
  Fellow 
  of 
  Trinity 
  

   College, 
  Cambridge. 
  Communicated 
  by 
  James 
  Glaisher, 
  

   F.R.S. 
  Received 
  November 
  23, 
  1874. 
  

  

  (Abstract.) 
  

  

  The 
  object 
  of 
  the 
  memoir 
  is 
  to 
  notice 
  certain 
  forms 
  into 
  which 
  the 
  

   primary 
  elliptic 
  functions 
  admit 
  of 
  being 
  thrown, 
  and 
  to 
  discuss 
  the 
  iden- 
  

   tical 
  relations 
  to 
  which 
  they 
  give 
  rise. 
  These 
  latter, 
  it 
  is 
  shown, 
  can 
  be 
  

   obtained 
  directly 
  by 
  the 
  aid 
  of 
  Fourier's 
  theorem, 
  or 
  in 
  a 
  less 
  straight- 
  

   forward 
  manner 
  by 
  ordinary 
  algebra. 
  

  

  Thus, 
  ex. 
  gr., 
  consider 
  the 
  cosine-amplitude 
  : 
  it 
  is 
  shown 
  that 
  we 
  have 
  

   the 
  formula 
  

  

  cosam 
  zKa?= 
  =— 
  , 
  \ 
  — 
  . 
  — 
  ; 
  

  

  , 
  1 
  + 
  1 
  fa, 
  I 
  

  

  ' 
  T 
  x— 
  2_J_ 
  r 
  — 
  (*— 
  2) 
  r 
  x+2^ 
  r 
  — 
  (x+2) 
  J 
  

  

  _ 
  ttK 
  

  

  (r 
  being 
  =e 
  K'), 
  which 
  is 
  noticeable 
  as 
  being 
  of 
  the 
  form 
  

  

  <])X 
  — 
  (jj(x 
  — 
  1) 
  — 
  <p(x+l) 
  + 
  &C, 
  

  

  and 
  therefore 
  an 
  analogue 
  of 
  

  

  1 
  1 
  1,1,1 
  6 
  

  

  7T 
  cosec 
  ttx=- 
  — 
  - 
  + 
  _ 
  + 
  — 
  _ 
  - 
  &c. 
  

  

  x 
  x 
  — 
  1 
  x-\-l 
  x 
  — 
  2 
  x 
  + 
  2 
  

  

  This 
  form 
  of 
  the 
  cosine-amplitude 
  gives 
  rise 
  to 
  the 
  identical 
  equation 
  

  

  sech.^— 
  sech 
  (x—fx)— 
  sech 
  (^H-/i)+ 
  sech 
  (x 
  — 
  2fx)-\- 
  sech 
  (# 
  + 
  2ju) 
  — 
  &c. 
  

  

  J2k_ 
  

  

  (sech 
  being 
  the 
  hyperbolic 
  secant). 
  The 
  result 
  (1) 
  is 
  deducible 
  at 
  once 
  

   from 
  the 
  integral 
  

  

  | 
  sech 
  cos 
  — 
  + 
  sech 
  ~ 
  C 
  os 
  ^~ 
  + 
  &c. 
  } 
  . 
  . 
  . 
  . 
  (1) 
  

  

  \ 
  sech 
  x 
  cos 
  nx 
  dx 
  = 
  ^ 
  sech 
  ^ 
  

  

  and 
  it 
  is 
  remarked 
  that 
  all 
  evaluable 
  integrals 
  of 
  the 
  forms 
  

  

  I 
  (f>x 
  cos 
  nx 
  dx, 
  I 
  \px 
  sin 
  nx 
  dx 
  

  

  