﻿ACADEMY 
  OF 
  SCIENCES. 
  99 
  

  

  ''^ 
  n. 
  n 
  71 
  

  

  XIV. 
  Having 
  given 
  the 
  sides 
  of 
  a 
  hollow 
  rectangle, 
  determine, 
  

   in 
  terms 
  of 
  those 
  sides, 
  the 
  sides 
  of 
  a 
  consecutive 
  series 
  of 
  similar 
  

   hollow 
  rectangles, 
  of 
  equal 
  areas 
  with 
  each 
  other, 
  into 
  which 
  it 
  is 
  

   required 
  to 
  divide 
  the 
  given 
  hollow 
  rectangle. 
  

  

  To 
  divide 
  it 
  into 
  n 
  hollow 
  rectangles, 
  let 
  I 
  ' 
  represent 
  the 
  outer 
  and 
  

   I 
  the 
  inner 
  lengths 
  of 
  the 
  given 
  hollow 
  rectangle 
  ; 
  b' 
  the 
  outer, 
  and 
  

   b 
  the 
  inner 
  breadths 
  ; 
  x, 
  ?/, 
  z, 
  etc., 
  the 
  consecutive 
  lengths, 
  reck- 
  

   oning 
  from 
  I' 
  to 
  I; 
  x 
  y\z\ 
  etc., 
  the 
  corresponding 
  consecutive 
  

   breadths 
  ; 
  {w 
  — 
  1) 
  and 
  w 
  the 
  last 
  two 
  lengths, 
  and 
  (w' 
  — 
  1) 
  and 
  

   w' 
  the 
  last 
  two 
  breadths 
  ; 
  then 
  

  

  y 
  I 
  n 
  p 
  V 
  I 
  n 
  ; 
  

  

  r_ 
  ^ 
  2hl+{n-2)h'l' 
  l 
  ,^^f/ 
  ^ 
  2bl 
  + 
  {n-2)b'r 
  l 
  . 
  ^^^ 
  

   ' 
  b' 
  X 
  n 
  S 
  r 
  I 
  n 
  i 
  • 
  • 
  

  

  (,,_lp- 
  l'\ 
  {n-2)bl 
  + 
  2yi' 
  l 
  ,__b' 
  Un-2)bl-^2h'l' 
  l 
  

  

  b' 
  I 
  n 
  y 
  I' 
  I 
  n 
  \ 
  

  

  XV. 
  Having 
  subdivided 
  the 
  hollow 
  rectangle, 
  as 
  in 
  problem 
  

   XIV., 
  determine, 
  in 
  terms 
  of 
  the 
  given 
  sides 
  thereof, 
  the 
  sides 
  of 
  

   a 
  consecutive 
  series 
  of 
  similar 
  hollow 
  rectangles 
  inside 
  the 
  given 
  

   rectangle, 
  and 
  having 
  areas 
  equal 
  with 
  those 
  of 
  the 
  prescribed 
  sub- 
  

   divisions. 
  

  

  Suppose 
  the 
  given 
  hollow 
  rectangle 
  is 
  subdivided 
  into 
  p 
  hollow 
  

   rectangles, 
  and 
  there 
  are 
  required 
  n 
  inner 
  hollow 
  rectangles 
  of 
  equal 
  

   areas 
  ; 
  let 
  Z', 
  Z, 
  6', 
  5, 
  x, 
  y, 
  3, 
  etc., 
  x\ 
  y\ 
  z', 
  etc., 
  represent 
  quan- 
  

   tities 
  as 
  before 
  ; 
  and 
  i\ 
  i", 
  ^"', 
  etc., 
  the 
  consecutive 
  lengths, 
  reck- 
  

   oning 
  from 
  I 
  toward 
  the 
  center; 
  i',i",i"',etc., 
  the 
  corresponding 
  

   consecutive 
  breadths 
  ; 
  then 
  

  

  