﻿4:66 
  Dr. 
  F. 
  L. 
  Hitchcock 
  on 
  the 
  Operator 
  V 
  w» 
  

  

  triple 
  and 
  the 
  double 
  integrals 
  over 
  the 
  volume 
  and 
  curved 
  sur- 
  

   face 
  of 
  the 
  hemisphere, 
  respectively. 
  In 
  employing 
  Gauss's 
  

   theorem, 
  however, 
  we 
  must 
  add 
  a 
  term 
  to 
  the 
  left 
  member 
  of 
  (2) 
  

   corresponding 
  to 
  integration 
  over 
  the 
  base 
  of 
  the 
  hemisphere. 
  

   If 
  the 
  base 
  is 
  taken 
  in 
  the 
  xy 
  plane 
  I 
  and 
  m 
  vanish 
  and 
  n 
  = 
  l. 
  

  

  If 
  Z= 
  -=r— 
  as 
  before, 
  the 
  term 
  to 
  be 
  added 
  to 
  the 
  left 
  of 
  the 
  

  

  oz 
  ( 
  PdF 
  

  

  equations 
  (2)-(4) 
  is 
  1 
  1 
  — 
  dA, 
  where 
  the 
  integration 
  is 
  over 
  

  

  a 
  circle 
  of 
  radius 
  a 
  in 
  the 
  xy 
  plane 
  with 
  centre 
  at 
  the 
  

   origin. 
  Eliminating 
  the 
  integration 
  over 
  the 
  curved 
  surface 
  

   as 
  before 
  we 
  now 
  have 
  

  

  Formula 
  (C) 
  

   ftj/(,) 
  . 
  W 
  m 
  (fi)dY 
  = 
  -L^^f^dr 
  . 
  [jJIfrfA 
  + 
  jtf 
  A 
  F 
  rfv] 
  

  

  • 
  • 
  • 
  (C) 
  

  

  where 
  the 
  triple 
  integrals 
  are 
  carried 
  over 
  the 
  volume 
  of 
  

   the 
  hemisphere, 
  the 
  double 
  integral 
  over 
  its 
  base, 
  the 
  latter 
  

   to 
  be 
  further 
  reduced 
  by 
  (B) 
  if 
  necessary. 
  

  

  As 
  an 
  elementary 
  example, 
  let 
  us 
  find 
  the 
  gravitational 
  

   moment 
  of 
  the 
  hemisphere 
  with 
  respect 
  to 
  its 
  base, 
  the 
  

   density 
  being 
  any 
  function 
  of 
  the 
  radius. 
  We 
  have 
  

  

  £[f/(r) 
  . 
  z 
  dV 
  = 
  ± 
  J 
  o 
  V/(r) 
  dr 
  . 
  [ 
  \\dk 
  + 
  0] 
  = 
  tt 
  J 
  V/(r) 
  dr. 
  

  

  Such 
  formulas 
  and 
  examples 
  can 
  be 
  multiplied 
  in 
  great 
  

   number. 
  Thus 
  by 
  quite 
  similar 
  reasoning 
  we 
  have 
  the 
  two- 
  

   dimensional 
  analogue 
  of 
  (C), 
  

  

  Formula 
  (D) 
  

  

  ...(D) 
  

  

  where 
  the 
  notation 
  is 
  as 
  in 
  (B), 
  and 
  the 
  double 
  integrals 
  are 
  

   over 
  a 
  semicircle 
  of 
  radius 
  a 
  and 
  centre 
  at 
  the 
  origin, 
  having 
  

   its 
  diameter 
  on 
  the 
  x 
  axis. 
  The 
  single 
  integral 
  is 
  taken 
  

   along 
  this 
  diameter, 
  that 
  is, 
  y 
  is 
  set 
  equal 
  to 
  zero 
  before 
  

   integrating. 
  

  

  These 
  formulas 
  may 
  be 
  extended 
  to 
  cover 
  cones, 
  cylinders, 
  

   segments 
  of 
  parabolas, 
  and 
  even 
  triangles. 
  As 
  a 
  final 
  

   example 
  of 
  the 
  sort 
  let 
  us 
  use 
  (D) 
  to 
  find 
  the 
  area 
  between 
  

   the 
  parabola 
  x 
  2 
  — 
  16#-f 
  4?/ 
  = 
  and 
  the 
  axis 
  of 
  x. 
  By 
  writing 
  

   \y 
  2 
  for 
  y 
  and 
  leaving 
  x 
  unchanged 
  we 
  carry 
  the 
  segment 
  of 
  

   the 
  parabola 
  into 
  a 
  semicircle. 
  We 
  need 
  not 
  move 
  the 
  origin, 
  

  

  