130        On  an  Instrument  for  Compounding  Vibrations. 
Note  on  the  Principle  of  the  Sand-  Clock. 
The  difficulty  of  propelling  a  column  of  sand,  occupying  a 
tube,  by  forces  pushing  at  one  end  is  well  known  ;  but  I  do  not 
remember  to  have  seen  any  discussion  of  the  question  on 
mechanical  principles.  A  similar  phenomenon  occurs  in  the 
storage  of  grain,  the  weight  of  which,  when  contained  in  tall 
bins,  is  found  to  be  taken  mainly  on  the  sides  and  but  little  on 
the  bottom  of  the  bin  *. 
The  unexpectedness  of  these  effects  depends  upon  a  half 
unconscious  comparison  with  fluids  which  in  a  state  of  rest 
are  exempt  from  friction.  In  the  present  case,  when  the  sand 
is  moving,  the  tangential  force  at  the  wall  is  reckoned  at  /x 
times  the  normal  force.  We  may  suppose,  as  a  rough 
approximation,  that  there  is  something  like  a  fluid  pressure  p. 
If  a  be  the  radius  of  the  tube  and  dx  an  element  of  length 
along  the  axis,  the  tangential  force  acting  upon  the  surface 
27TA  dx  is  fip .  2ira  .  dx.  This  is  to  be  equated  to  the  difference 
of  the  forces  upon  the  two  faces  of  the  slice,  viz.  —  7ra2dp. 
Accordingly 
dp     2yLt 
pdx  a 
or 
p  =  p0e-2^\ 
the  pressure  diminishing  as  a  increases.  Hence  a  powerful 
pressure  at  #  =  0  is  unable  to  overcome  a  very  feeble  one 
acting  in  the  opposite  direction  at  a  section  many  diameters 
away.  The  case  is  similar  to  that  of  a  rope  coiled  round  a 
post,  as  used  to  check  the  motion  of  steamers  coming  up  to  a 
pier. 
As  regards  numbers,  it  will  not  be  out  of  the  way  to  suppose 
^  =  tV-     When  .r=10a,  jt)/p0  =  ^~2  =  *14. 
Nov.  22,  1905. 
*  I.  Roberts,  Proc.  Roy.  Soc.  vol.  xxxvi.  p.  226,  1884.  "In  any 
cell  which  has  parallel  sides,  the  pressure  of  wheat  upon  the  bottom 
ceases  when  it  is  charged  up  to  twice  the  diameter  of  the  inscribed 
circle." 
