﻿120 
  Dr. 
  J. 
  G. 
  Leathern 
  on 
  the 
  

  

  2. 
  The 
  geometrical, 
  or 
  (z, 
  ?), 
  relation 
  may 
  be 
  either 
  in 
  the 
  

   form 
  

  

  *=/(«, 
  (i) 
  

  

  or 
  in 
  the 
  differential 
  form 
  

  

  <fe=^f(ry?, 
  (2) 
  

  

  where 
  &(£) 
  is 
  a 
  periodic 
  curve-factor 
  of 
  linear 
  period 
  X 
  and 
  

   angular 
  period 
  2tt. 
  With 
  this 
  angular 
  period 
  it 
  is 
  necessary 
  * 
  

   that 
  both 
  £f 
  (?) 
  and 
  /(?) 
  should, 
  for 
  rj 
  great 
  and 
  positive, 
  

   tend 
  to 
  infinity 
  like 
  exp 
  (%Trrj/\). 
  In 
  fact, 
  £f(?) 
  is 
  expan- 
  

   sible 
  in 
  the 
  form 
  

  

  £f(?) 
  = 
  exp 
  (-2m?/\) 
  . 
  2c*exp 
  (2tto?/\), 
  . 
  (3) 
  

  

  where 
  5 
  = 
  0, 
  2, 
  3, 
  4 
  ... 
  , 
  and 
  the 
  coefficients 
  may 
  be 
  complex. 
  

   The 
  periodicity 
  of 
  z 
  makes 
  it 
  necessary 
  f 
  that 
  ^ 
  = 
  0. 
  

   From 
  integration 
  of 
  (3) 
  it 
  follows 
  that 
  

  

  *=/(?) 
  =c 
  + 
  (t\/27r) 
  exp(-27n'?/A,) 
  . 
  2{c«/(l-«) 
  } 
  exp 
  (27m?/\), 
  (4) 
  

  

  where 
  c 
  is 
  another 
  complex 
  constant 
  J. 
  

  

  It 
  is 
  also 
  to 
  be 
  noticed 
  that, 
  if 
  | 
  <^(?) 
  | 
  = 
  h, 
  and 
  if 
  

   ^ 
  = 
  K,exp(; 
  7s 
  ), 
  

  

  h? 
  = 
  exp 
  UjjS 
  j 
  Kq 
  2 
  + 
  exp 
  ( 
  p 
  J 
  2k 
  k 
  2 
  cos 
  (^ 
  -^ 
  + 
  y 
  2 
  - 
  7o 
  j 
  

  

  + 
  exp 
  ( 
  — 
  -^p) 
  2 
  *o*3 
  cos 
  (-^- 
  +7s-7o) 
  

  

  + 
  exp^-^J<| 
  2/t 
  «: 
  4 
  cos^-^+74-7oj 
  + 
  «: 
  2 
  2 
  |... 
  , 
  . 
  (5) 
  

  

  the 
  terms 
  containing 
  ascending 
  integral 
  powers 
  of 
  

   exp 
  (~27rrj/\). 
  

  

  * 
  L. 
  c. 
  § 
  5. 
  

  

  ti.c§4. 
  

  

  J 
  The 
  problem 
  of 
  obtaining 
  a 
  transformation 
  of 
  the 
  type 
  of 
  formula 
  (4), 
  

   ao 
  that 
  a 
  given 
  closed 
  curve 
  shall 
  correspond 
  to 
  rj=0, 
  is 
  the 
  same 
  as 
  that 
  

   of 
  the 
  parametric 
  representation 
  of 
  the 
  given 
  curve 
  by 
  a 
  formula 
  of 
  the 
  

   type 
  

  

  oc 
  + 
  iy 
  = 
  m 
  exp 
  ( 
  - 
  if) 
  +n 
  + 
  2 
  ^ 
  exp 
  (isf) 
  , 
  

  

  where 
  <p 
  is 
  a 
  real 
  parameter, 
  m 
  and 
  the 
  /u's 
  are 
  complex 
  constants, 
  and 
  s 
  

   takes 
  positive 
  integral 
  values. 
  

  

  It 
  is 
  to 
  be 
  noted 
  that 
  a 
  formula 
  of 
  this 
  type 
  need 
  not 
  represent 
  a 
  curve 
  

   free 
  from 
  nodes 
  unless 
  the 
  constants 
  are 
  suitably 
  restricted. 
  

  

  