﻿198 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  

  

  v 
  cos 
  ifr. 
  Hence 
  in 
  a 
  unit 
  of 
  times 
  \nv 
  cos 
  yjr 
  sin 
  ^cty 
  comets 
  

   will 
  cross 
  a 
  nnit 
  of 
  the 
  surface 
  % 
  gomg 
  towards 
  the 
  sun. 
  

   The 
  total 
  entering 
  the 
  sphere 
  in 
  the 
  unit 
  of 
  time 
  will 
  be 
  this 
  

   number 
  multiplied 
  by 
  the 
  number 
  of 
  units 
  in 
  the 
  surface 
  of 
  

   0, 
  or 
  

  

  7T 
  

  

  \7tr 
  z 
  I 
  \ 
  nv 
  cos 
  ip 
  sin 
  ip 
  dip 
  = 
  nnvr 
  1 
  . 
  

  

  Fig. 
  14; 
  u=70. 
  

  

  —■ 
  -^ 
  

  

  X 
  

  

  

  X 
  

  

  n 
  

  

  ! 
  

  

  

  ^_ 
  /_ 
  

  

  

  V- 
  J- 
  

  

  ^) 
  A 
  

  

  It 
  y 
  

  

  )_^Or* 
  

  

  1 
  

  

  Fig. 
  15; 
  u=110. 
  

   H 
  

  

  

  

  

  

  

  

  \ 
  

  

  

  

  

  \ 
  

  

  

  

  

  \ 
  

  

  

  

  

  

  

  X 
  

  

  

  i 
  

  

  

  

  / 
  

  

  

  

  ) 
  

  

  / 
  

  

  

  } 
  

  

  Y 
  

  

  

  

  f 
  

  

  

  || 
  

  

  28. 
  Distribution 
  of 
  parabolic 
  comets 
  as 
  to 
  perihelion 
  dis- 
  

   tance. 
  — 
  This 
  supposition 
  of 
  equable 
  distribution 
  of 
  the 
  goals 
  

   of 
  comets 
  as 
  they 
  cross 
  the 
  spherical 
  surface 
  5 
  involves 
  also 
  a 
  

   law 
  of 
  distribution 
  of 
  comets 
  as 
  to 
  perihelion 
  distance. 
  The 
  

   number 
  of 
  comets 
  that 
  enter 
  the 
  sphere 
  in 
  a 
  given 
  time 
  whose 
  

   motions 
  make 
  with 
  the 
  normal 
  angles 
  between 
  yfr 
  and 
  i/r 
  + 
  d^r 
  

   is 
  proportional 
  to 
  sin 
  ^ 
  cos 
  ^d^jr. 
  If 
  1ST 
  be 
  the 
  number 
  of 
  

   comets 
  that 
  enter 
  in 
  a 
  given 
  period 
  of 
  time 
  with 
  an 
  angle 
  with 
  

   the 
  normal 
  less 
  than 
  -\fr, 
  we 
  may 
  write 
  dl$=k 
  sin 
  yjr 
  cos 
  yjrdyjr, 
  

   where 
  k 
  is 
  some 
  constant. 
  But 
  if 
  q 
  is 
  the 
  perihelion 
  distance 
  

   of 
  a 
  comet 
  which 
  at 
  the 
  distance 
  r 
  from 
  the 
  sun 
  moves 
  at 
  an 
  

   angle 
  with 
  the 
  radius 
  equal 
  to 
  i/r, 
  then 
  q—r 
  sin 
  2 
  yjr, 
  and 
  dq=2r 
  

   sin 
  yjr 
  cos 
  -^rd^fr. 
  But 
  comets 
  that 
  enter 
  Q 
  with 
  angles 
  to 
  the 
  

  

  