﻿Cloud 
  of 
  similar 
  small 
  Particles 
  of 
  any 
  Shape, 
  

  

  377 
  

  

  represent 
  the 
  inclination 
  of 
  "WU 
  to 
  ZW 
  produced, 
  fig. 
  2. 
  

   The 
  direction 
  cosines 
  of 
  either 
  set 
  of 
  axes 
  with 
  respect 
  to 
  

  

  Fig. 
  2. 
  

  

  the 
  other 
  are 
  given 
  by 
  the 
  formulae 
  * 
  

  

  cos 
  XU 
  = 
  — 
  sin 
  (/> 
  sin 
  a/t 
  + 
  cos 
  <j> 
  cos-»/r 
  cos 
  

   cos 
  YU 
  = 
  cos 
  (f> 
  sin 
  -v/r-fsinc£ 
  cos-^ 
  cos 
  6 
  \ 
  

   cos 
  ZU 
  = 
  — 
  sin 
  cos 
  yjr 
  

  

  cos 
  XV 
  = 
  —sin 
  <f> 
  cos 
  ty 
  — 
  cos 
  <j> 
  sin 
  ^ 
  cos 
  6 
  

   cos 
  YV 
  = 
  cos 
  <£ 
  cos 
  y 
  — 
  sin 
  <£ 
  sin 
  i/r 
  cos 
  6 
  

   sin 
  # 
  sin 
  yjr 
  

  

  cosZV 
  = 
  

  

  cosXW 
  = 
  

   cosYW 
  = 
  

   cosZW 
  = 
  

  

  (7) 
  

  

  («) 
  

  

  sin 
  6 
  cos 
  (/> 
  

  

  sin 
  sin 
  </> 
  j* 
  (9j 
  

  

  J 
  

  

  cos 
  

  

  Supposing, 
  as 
  before, 
  that 
  the 
  primary 
  vibration 
  is 
  parallel 
  

   to 
  Z, 
  we 
  have 
  as 
  the 
  first 
  set 
  of 
  factors 
  

  

  cos 
  ZU 
  = 
  —sin 
  6 
  cos 
  \/r, 
  

   cos 
  ZV 
  = 
  sin 
  6 
  sin 
  ^ 
  

   cos 
  Z 
  W 
  = 
  cos 
  

  

  1 
  

  

  (10) 
  

  

  For 
  the 
  vibrations 
  propagated 
  along 
  OY 
  which 
  are 
  parallel 
  

   to 
  Z, 
  we 
  have 
  the 
  same 
  factors 
  over 
  again 
  with 
  coefficients 
  

  

  * 
  See, 
  for 
  example, 
  Routh's 
  * 
  Kigid 
  Dynamics,' 
  Part 
  I. 
  § 
  258, 
  1897. 
  

   \\r 
  and 
  <p 
  are 
  interchanged. 
  

  

  