﻿154 
  Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fermafs 
  Law. 
  

  

  parallel 
  to 
  the 
  axes, 
  and 
  Z, 
  m, 
  n 
  the 
  direction-cosines 
  oE 
  an 
  

   element 
  of 
  surface. 
  

  

  Then, 
  since 
  in 
  any 
  limited 
  region 
  the 
  total 
  number 
  must 
  

   remain 
  unchanged, 
  

  

  ^{ff+mg 
  + 
  nK)d$ 
  = 
  0== 
  ^(k^S 
  (say) 
  

  

  — 
  I 
  (If 
  -f 
  mg' 
  4- 
  nh')d$ 
  + 
  terms 
  depending 
  on 
  the 
  motion 
  

   * 
  of 
  thei 
  tubes, 
  

  

  [/'■f. 
  *4 
  

  

  Suppose 
  the 
  tubes 
  across 
  ^S 
  spread 
  over 
  dS' 
  on 
  account 
  of 
  

   this 
  velocity, 
  then 
  the 
  terms 
  depending 
  on 
  the 
  velocity 
  

  

  =j'(R 
  n 
  / 
  JS 
  , 
  -R^S). 
  

  

  Consider 
  the 
  volume 
  enclosed 
  by 
  d$ 
  and 
  dS'. 
  The 
  total 
  

   surface 
  integral 
  over 
  this 
  surface 
  

  

  But 
  this 
  must 
  be 
  

  

  = 
  — 
  'Rf/dS' 
  + 
  E 
  w 
  <iS 
  4- 
  surface 
  integrals 
  over 
  the 
  tubular 
  

   surface 
  generated 
  by 
  the 
  contour 
  of 
  rfS. 
  

  

  Now, 
  if 
  ds=a,n 
  element 
  of 
  arc 
  of 
  dS 
  (at 
  a?, 
  y, 
  z), 
  

  

  U 
  = 
  velocity 
  of 
  a 
  tube, 
  so 
  that 
  

  

  \Jdt 
  = 
  length 
  of 
  the 
  line 
  joining 
  corresponding 
  points 
  

   of 
  c?S 
  and 
  dS 
  x 
  , 
  

  

  e 
  = 
  inclination 
  of 
  ds 
  to 
  the 
  direction 
  of 
  the 
  dis- 
  

   placement 
  of 
  a 
  tube 
  ; 
  

  

  then, 
  ds 
  JJdt 
  sin 
  e 
  = 
  area 
  of 
  the 
  portion 
  of 
  the 
  tubular 
  surface 
  

   generated 
  by 
  ds. 
  

  

  Then, 
  the 
  surface 
  integral 
  over 
  the 
  tubular 
  surface 
  

  

  = 
  f 
  H 
  n 
  ds 
  dt 
  U 
  sin 
  e 
  

  

  dt 
  

  

  dx 
  dy 
  

  

  dz 
  

  

  f 
  9 
  

  

  h 
  

  

  U 
  V 
  

  

  w 
  

  

  where 
  u, 
  v, 
  w 
  are 
  the 
  component 
  velocities 
  of 
  a 
  tube 
  corre- 
  

   sponding 
  to 
  the 
  velocity 
  U, 
  

  

  = 
  (Ldx 
  + 
  Mdy 
  + 
  Ndz)dt 
  (say) 
  

   where 
  

  

  L, 
  M, 
  N=(gw 
  — 
  vh), 
  (uh-fw), 
  (fv-ug). 
  

  

  