﻿288 
  Prof. 
  A. 
  Anderson 
  on 
  the 
  Coefficients 
  of 
  

  

  where 
  G 
  7 
  and 
  F, 
  are 
  the 
  sums 
  of 
  the 
  two 
  series, 
  the 
  sub- 
  

   script^' 
  denoting 
  that 
  J's 
  enter 
  into 
  the 
  expressions. 
  

   In 
  like 
  manner, 
  the 
  second 
  set 
  gives 
  immediately 
  

  

  -*(l4aV)[^ 
  + 
  AIi 
  .M..BI. 
  + 
  -]' 
  

  

  or 
  cY 
  = 
  bJJGi-b(l 
  + 
  aV)Fi. 
  

  

  Now 
  pu 
  = 
  Y 
  + 
  - 
  , 
  and 
  pi2 
  = 
  ~U 
  , 
  

  

  hence 
  (c 
  + 
  abY^p^ 
  — 
  aGjpn, 
  

  

  and 
  clp 
  n 
  \+abFip 
  X 
  i 
  = 
  b(!xipi<t. 
  

  

  Hence 
  ., 
  ab-^ 
  

  

  1 
  c 
  J 
  

  

  Pn 
  

  

  Pn 
  

  

  1 
  % 
  

  

  _ 
  1 
  c 
  

  

  (i+^)(i+^)-^; 
  

  

  Since, 
  however, 
  jp 
  12 
  must 
  be 
  equal 
  to 
  p 
  2U 
  it 
  follows 
  that 
  

   Gr,-=G,-, 
  and, 
  in 
  fact, 
  on 
  examining 
  the 
  two 
  series, 
  it 
  will 
  be 
  

   seen 
  that 
  they 
  are 
  identical 
  if 
  

  

  BJ 
  1 
  .AJ 
  2 
  =AT 
  1 
  .BI 
  2> 
  

  

  BJ 
  3 
  .AJ 
  4 
  =AI 
  3 
  .BI 
  4 
  , 
  

   &c. 
  = 
  &c. 
  

  

  These 
  equalities, 
  though 
  they 
  can 
  be 
  proved 
  without 
  

   difficulty, 
  are 
  not 
  at 
  all 
  self-evident, 
  and 
  imply 
  the 
  further 
  

   relations 
  

  

  AT 
  1 
  -BJ 
  1 
  -Al3-BJ 
  3 
  =...==AI 
  2n+1 
  -BJ 
  2n+1 
  =...-^^ 
  2 
  . 
  

  

  c 
  

  

  Thus 
  we 
  have 
  a 
  case 
  where 
  theorems 
  in 
  pure 
  geometry 
  

   are 
  suggested 
  by 
  purely 
  electrical 
  ones. 
  

  

  