﻿300 
  

  

  On 
  the 
  Value 
  of 
  a 
  certain 
  Definite 
  Integral. 
  [Mar. 
  4 
  

  

  II. 
  Semidiurnal 
  Tide 
  . 
  

  

  Lunar 
  Semidiurnal 
  Tide. 
  

  

  True 
  Lunitidal 
  Interval 
  23 
  h 
  48 
  m 
  I 
  s 
  . 
  

  

  Uncorrected 
  ratio 
  of 
  Solar 
  and 
  1 
  S" 
  _ 
  f 
  0*412 
  inch 
  (height). 
  

   Lunar 
  Coefficients 
  J 
  W' 
  = 
  { 
  0-549 
  „ 
  (time). 
  

  

  II. 
  "Note 
  on 
  the 
  Value 
  of 
  a 
  certain 
  Definite 
  Integral." 
  By 
  

   I. 
  Todhunter, 
  M.A., 
  F.R.S., 
  Honorary 
  Fellow 
  of 
  St. 
  John's 
  

   College, 
  Cambridge. 
  Received 
  February 
  13, 
  1875. 
  

  

  Let 
  P 
  m 
  (#) 
  denote 
  Legendre's 
  coefficient 
  of 
  the 
  order 
  m, 
  and 
  P 
  n 
  (x) 
  that 
  

  

  of 
  the 
  order 
  n 
  : 
  it 
  is 
  required 
  to 
  find 
  the 
  value 
  of 
  I 
  P 
  m 
  (#) 
  P 
  n 
  (x)dx. 
  We 
  

  

  need 
  not 
  consider 
  the 
  case 
  in 
  which 
  m=n; 
  for 
  it 
  is 
  an 
  established 
  result 
  

   that 
  the 
  value 
  of 
  the 
  integral 
  taken 
  between 
  the 
  limits 
  — 
  1 
  and 
  1 
  is 
  then 
  

   2 
  

  

  equal 
  to 
  2 
  n 
  -\-V 
  an( 
  ^ 
  ^ 
  e 
  vauie 
  De 
  ^ 
  ween 
  the 
  limits 
  and 
  1 
  will 
  be 
  half 
  of 
  

   this. 
  We 
  suppose 
  now 
  that 
  m 
  and 
  n 
  are 
  different. 
  

   It 
  is 
  known 
  that 
  

  

  n 
  , 
  ^ 
  1.3.5.. 
  (2m- 
  1) 
  f 
  m 
  m(m— 
  1) 
  , 
  "1 
  

  

  (1) 
  

  

  and 
  also 
  that 
  

  

  1 
  J^ 
  1 
  - 
  r2 
  >^^} 
  +'»(« 
  + 
  !) 
  P.(*)=0 
  (2) 
  

  

  dx 
  

  

  Similar 
  expressions, 
  of 
  course, 
  hold 
  for 
  ¥ 
  n 
  {x). 
  

  

  Now, 
  by 
  a 
  well-known 
  process 
  of 
  integration 
  by 
  parts, 
  we 
  deduce 
  

  

  {m(m 
  + 
  1) 
  - 
  n(n 
  + 
  1)} 
  J 
  VJsc) 
  V 
  n 
  {x)dx 
  

  

  this 
  formula 
  can 
  also 
  be 
  immediately 
  verified 
  by 
  differentiation 
  and 
  the 
  

  

  use 
  of 
  (2). 
  

  

  Thus 
  

  

  { 
  m 
  ( 
  m 
  + 
  l)-n(n+ 
  1)} 
  J* 
  PjV) 
  V 
  n 
  (x)dx 
  

  

  If 
  m 
  and 
  n 
  are 
  both 
  even 
  or 
  both 
  odd, 
  the 
  right-hand 
  member 
  of 
  the 
  

   last 
  formula 
  will 
  vanish 
  by 
  (1) 
  ; 
  thus 
  the 
  only 
  case 
  we 
  have 
  to 
  consider 
  

  

  