﻿ACADEMY 
  OP 
  SCIENCES. 
  5 
  

  

  cysticircus, 
  or 
  imperfectly 
  developed 
  worm, 
  as 
  it 
  is 
  found 
  in 
  the 
  

   animal 
  tissues. 
  It 
  appears 
  to 
  hold 
  a 
  middle 
  place 
  between 
  the 
  

   cysticircus 
  and 
  perfect 
  worm 
  as 
  found 
  in 
  the 
  intestines. 
  

  

  New 
  Problems 
  in 
  Mensuration.* 
  

  

  BY 
  GEORGE 
  DAVIDSON. 
  

  

  9. 
  Given 
  the 
  diameter 
  of 
  a 
  sphere, 
  to 
  find 
  the 
  diameters 
  of 
  any 
  number 
  of 
  

   concentric 
  shells 
  and 
  the 
  central 
  sphere 
  having 
  equal 
  volumes 
  with 
  each 
  other, 
  

   into 
  which 
  the 
  given 
  sphere 
  is 
  divided. 
  

  

  To 
  divide 
  it 
  into 
  n 
  shells 
  and 
  the 
  inner 
  sphere, 
  let 
  d 
  represent 
  the 
  diameter 
  of 
  

   the 
  given 
  sphere 
  ; 
  x, 
  y, 
  z, 
  etc., 
  the 
  diameters 
  next 
  interior 
  ; 
  (w-l) 
  and 
  w 
  the 
  

   diameters 
  of 
  last 
  shell 
  and 
  inner 
  sphere 
  ; 
  then, 
  

  

  x» 
  = 
  ^i 
  '- 
  : 
  ««J 
  = 
  1 
  '. 
  etc. 
  

  

  (w=l)3 
  = 
  ^'^^ 
  • 
  w3 
  = 
  '^^ 
  

  

  11 
  n 
  

  

  10. 
  Given 
  the 
  diameters 
  of 
  a 
  spherical 
  shell, 
  to 
  find 
  the 
  diameters 
  of 
  any 
  num- 
  

   ber 
  of 
  concentric 
  shells 
  into 
  which 
  it 
  is 
  divided, 
  they 
  having 
  equal 
  volumes 
  with 
  

   each 
  other. 
  

  

  To 
  divide 
  it 
  into 
  p 
  shells, 
  let 
  d' 
  represent 
  the 
  exterior 
  diameter 
  ; 
  d 
  the 
  inter- 
  

   ior 
  diameter 
  ; 
  x, 
  y, 
  z, 
  etc., 
  the 
  diameters 
  reckoned 
  from 
  d' 
  towards 
  d 
  ; 
  (w-1) 
  

   and 
  w 
  the 
  last 
  two 
  diameters 
  ; 
  then, 
  

  

  etc. 
  

  

  xS 
  = 
  c/3 
  + 
  jp-l) 
  (1/2 
  . 
  ^3 
  ^ 
  2(IS+(p-2) 
  d'3 
  . 
  

  

  P 
  ' 
  P 
  ' 
  

  

  (,„_1).-^- 
  (p-2)(l3 
  + 
  2d'3 
  . 
  ,„3_ 
  {p-l) 
  (13 
  + 
  (1'3 
  

  

  P 
  P 
  

  

  11. 
  Given 
  the 
  diameters 
  of 
  a 
  spherical 
  shell, 
  and 
  having 
  divided 
  it 
  into 
  any 
  

   number 
  of 
  concentric 
  shells 
  of 
  equal 
  volumes 
  with 
  each 
  other, 
  to 
  find 
  the 
  diam- 
  

   eters 
  of 
  the 
  consecutive 
  outer 
  shells 
  of 
  equal 
  volumes 
  with 
  the 
  subdivided 
  shell. 
  

  

  Let 
  d',d; 
  a:, 
  J/, 
  etc., 
  represent 
  the 
  quantities 
  as 
  before; 
  p 
  the 
  number 
  of 
  

   shells 
  into 
  which 
  the 
  given 
  shell 
  is 
  divided 
  ; 
  n 
  the 
  required 
  number 
  of 
  outer 
  

   shells, 
  and 
  o', 
  o", 
  o'", 
  etc., 
  the 
  consecutive 
  diameters 
  of 
  these 
  shells 
  ; 
  then, 
  

  

  3 
  _ 
  (n 
  + 
  p) 
  d'3 
  — 
  nd3 
  

   ^ 
  n 
  p 
  

  

  12. 
  Given 
  the 
  diameter 
  of 
  a 
  spherical 
  shell, 
  and 
  having 
  divided 
  it 
  into 
  any 
  

  

  * 
  Tn 
  continuation 
  of 
  former 
  problems 
  ; 
  vide 
  Vol. 
  IV, 
  pp. 
  278, 
  290. 
  Note. 
  — 
  

   Please 
  note 
  the 
  following 
  typographical 
  errors 
  in 
  said 
  problems 
  : 
  Page 
  290, 
  

   second 
  line 
  from 
  bottom, 
  for 
  "vi2," 
  read"a:2."; 
  page 
  290, 
  last 
  line, 
  for 
  " 
  t<;," 
  

   read 
  " 
  w^." 
  

  

  