﻿420 
  H. 
  A. 
  Newton 
  — 
  Plane 
  of 
  Jupiter* 
  s 
  orbit, 
  etc. 
  

  

  Art. 
  XXXIV. 
  — 
  Relation 
  of 
  the 
  plane 
  of 
  Jupiter 
  ] 
  s 
  orbit 
  to 
  the 
  

   mean-plane 
  of 
  four 
  hundred 
  and 
  one 
  minor 
  planet 
  orbits 
  : 
  

   by 
  H. 
  A. 
  Newton. 
  

  

  About 
  nine 
  years 
  ago 
  (this 
  Journal, 
  III, 
  xxxi, 
  p. 
  319) 
  I 
  

   called 
  attention 
  in 
  a 
  brief 
  note 
  to 
  the 
  fact 
  that 
  the 
  mean- 
  plane 
  

   of 
  the 
  orbits 
  of 
  the 
  then 
  known 
  two 
  hundred 
  and 
  fifty-one 
  

   minor 
  planets 
  was 
  inclined 
  to 
  the 
  plane 
  of 
  Jupiter's 
  orbit 
  by 
  a 
  

   very 
  small 
  angle. 
  In 
  fact 
  no 
  minor 
  planet 
  out 
  of 
  the 
  whole 
  

   251 
  had 
  its 
  plane 
  so 
  near 
  to 
  the 
  mean-plane 
  as 
  did 
  the 
  

   planet 
  Jupiter. 
  Since 
  that 
  time 
  we 
  have 
  added 
  to 
  the 
  list 
  of 
  

   planets 
  between 
  Mars 
  and 
  Jupiter 
  one 
  hundred 
  and 
  fifty 
  newly 
  

   discovered 
  ones, 
  and 
  it 
  seems 
  worth 
  while 
  to 
  find 
  whether 
  the 
  

   same 
  relation 
  of 
  the 
  large 
  planet 
  to 
  the 
  entire 
  group 
  of 
  four 
  

   hundred 
  and 
  one 
  small 
  ones 
  holds 
  true. 
  

  

  The 
  plane 
  of 
  an 
  orbit 
  is 
  determined 
  by 
  the 
  longitude 
  of 
  the 
  

   ascending 
  node 
  and 
  the 
  inclination, 
  and 
  its 
  place 
  may 
  be 
  repre- 
  

   sented 
  to 
  the 
  eye 
  by 
  a 
  plot 
  in 
  which 
  the 
  inclination 
  is 
  the 
  

   radius 
  vector 
  and 
  the 
  longitude 
  of 
  the 
  node 
  is 
  the 
  polar 
  angle. 
  

   The 
  point 
  thus 
  plotted 
  is 
  of 
  course 
  the 
  pole 
  of 
  the 
  plane. 
  

  

  The 
  mean-plane 
  of 
  the 
  401 
  planes, 
  regarding 
  each 
  plane 
  as 
  a 
  

   unit, 
  may 
  be 
  determined 
  with 
  sufficient 
  accuracy 
  for 
  the 
  

   present 
  purpose 
  by 
  the 
  formulas 
  for 
  computing 
  the 
  center 
  of 
  

   gravity 
  of 
  the 
  401 
  points 
  plotted, 
  viz 
  : 
  

  

  40 
  1 
  / 
  cos 
  LI 
  = 
  2 
  i 
  cos 
  Q 
  , 
  and 
  401 
  / 
  sin 
  £1 
  = 
  2i 
  sin 
  Q> 
  : 
  

  

  where 
  i 
  and 
  9> 
  are 
  the 
  inclination 
  and 
  longitude 
  of 
  ascending 
  

   node 
  of 
  any 
  orbit, 
  and 
  I 
  and 
  Q 
  are 
  the 
  same 
  functions 
  of 
  the 
  

   mean-plane 
  of 
  all 
  the 
  orbits. 
  Computing 
  I 
  and 
  & 
  for 
  the 
  401 
  

   orbits 
  as 
  given 
  in 
  the 
  Annuaire 
  du 
  Bureau 
  des 
  Longitudes 
  

   for 
  1895, 
  adding 
  three 
  later 
  orbits 
  from 
  the 
  Astronomische 
  

   Nachriohten, 
  we 
  have 
  

  

  7=0°-93, 
  and 
  £1 
  = 
  109°-3. 
  

  

  The 
  corresponding 
  quantities 
  for 
  Jupiter 
  are 
  

  

  i~ 
  1°'31, 
  Q 
  = 
  98°*9, 
  

  

  so 
  that 
  the 
  inclination 
  of 
  the 
  mean 
  plane 
  to 
  Jupiter's 
  plane 
  is 
  

   o, 
  43. 
  The 
  three 
  minor 
  planets 
  whose 
  planes 
  are 
  nearest 
  to 
  

   the 
  mean-plane 
  are 
  1893 
  Y, 
  (27) 
  and 
  (149). 
  These 
  planes 
  

   make 
  angles 
  with 
  the 
  mean-plane 
  severally 
  equal 
  to 
  0°*65, 
  

   0°-74, 
  and 
  0°'77. 
  The 
  planet 
  1893 
  Y, 
  was 
  photographically 
  

   discovered 
  and 
  has 
  not 
  yet 
  a 
  place 
  in 
  the 
  numbered 
  series 
  of 
  

   planets. 
  Its 
  plane 
  will 
  doubtless 
  be 
  much 
  changed 
  when 
  the 
  

  

  