﻿496 
  Mr. 
  Nalinimohan 
  Basu 
  on 
  

  

  great 
  increase 
  in 
  the 
  bowing 
  pressure 
  which 
  becomes 
  

   necessary 
  at 
  the 
  wol£-note 
  pitch, 
  and 
  prove 
  that 
  the 
  effect 
  

   of 
  the 
  mute 
  is 
  to 
  increase 
  the 
  bowing 
  pressure 
  necessary 
  at 
  

   low 
  frequencies 
  and 
  to 
  decrease 
  it 
  at 
  high 
  frequencies. 
  With 
  

   this 
  mechanical 
  player, 
  the 
  " 
  cyclical 
  " 
  or 
  " 
  beating 
  " 
  tones 
  

   obtained 
  in 
  certain 
  cases 
  (Phil. 
  Mag. 
  Oct. 
  1916 
  and 
  Feb. 
  1917) 
  

   may 
  be 
  steadily 
  maintained 
  and 
  controlled 
  by 
  suitable 
  adjust- 
  

   ment 
  of 
  the 
  bowing 
  pressure. 
  

   Indian 
  Association 
  for 
  the 
  

   Cultivation 
  of 
  Science, 
  

   Calcutta. 
  

  

  LX. 
  On 
  a 
  New 
  Type 
  of 
  Rough 
  Surface 
  the 
  Motion 
  of 
  a 
  

   Heavy 
  Particle 
  on 
  which 
  is 
  determinable 
  by 
  Quadratures. 
  

   By 
  Nalinimohan 
  Basu, 
  Af.Sc, 
  University 
  Lecturer 
  in 
  

   Applied 
  Mathematics, 
  Calcutta*. 
  

  

  1. 
  TT 
  is 
  well 
  known 
  that 
  the 
  motion 
  of 
  a 
  heavy 
  particle 
  on 
  

   JL 
  a 
  rough 
  surface 
  is 
  determinable 
  by 
  quadratures 
  when 
  

   the 
  surface 
  is 
  an 
  inclined 
  plane, 
  a 
  circular 
  cylinder, 
  a 
  circular 
  

   cone, 
  or 
  a 
  vertical 
  cylinder 
  standing 
  on 
  a 
  logarithmic 
  spiral 
  

   as 
  the 
  base 
  f. 
  

  

  The 
  object 
  of 
  the 
  present 
  paper 
  is 
  to 
  make 
  known 
  another 
  

   surface 
  which 
  has, 
  in 
  a 
  certain 
  sense, 
  the 
  same 
  property 
  as 
  

   the 
  four 
  surfaces 
  mentioned 
  above 
  and 
  which, 
  it 
  is 
  believed, 
  

   has 
  not 
  been 
  considered 
  by 
  any 
  previous 
  writer. 
  

  

  2. 
  Let 
  us 
  consider 
  the 
  surface 
  whose 
  equation 
  is 
  

  

  % 
  = 
  2~tana.tan- 
  1 
  ^-(^ 
  2 
  + 
  2/ 
  2 
  -l)P(^)=0, 
  . 
  (1) 
  

  

  where 
  P(V) 
  is 
  given 
  by 
  the 
  relation 
  

  

  6 
  

  

  2ft 
  cot 
  « 
  cos 
  a 
  

  

  log 
  

  

  1 
  + 
  tan 
  

  

  a 
  

  

  l_ 
  tan 
  « 
  VW'cof 
  

   2 
  

  

  6 
  

  

  Xlog 
  

  

  \/l 
  — 
  /iicota-\- 
  \ 
  ; 
  1 
  -\- 
  jju 
  cot 
  a 
  .tan 
  

  

  ^l—fxcota— 
  s/ 
  1 
  ■{- 
  p 
  cot 
  a 
  . 
  tan 
  t, 
  

  

  z-h, 
  . 
  (2) 
  

  

  tan 
  standing 
  for 
  2P 
  cos 
  u 
  and 
  the 
  axis 
  of 
  z 
  being 
  drawn 
  

   vertically 
  upwards. 
  

  

  * 
  Communicated 
  by 
  Prof. 
  G. 
  Prasad. 
  

  

  t 
  For 
  the 
  first 
  three 
  cases, 
  see 
  any 
  well 
  known 
  text-book 
  on 
  the 
  

   " 
  Dynamics 
  of 
  a 
  Particle," 
  e. 
  g. 
  Routh's 
  book 
  ; 
  for 
  the 
  last 
  case, 
  see 
  

   A. 
  Razzaboni's 
  paper, 
  " 
  Sul 
  movimento 
  d'un 
  punto 
  materiale 
  sopra 
  una 
  

   superficie 
  non 
  levigata." 
  (Giornale 
  di 
  matematiche, 
  vol. 
  xxxiv.) 
  

  

  