6    For  the  case  of   the  balls,  chiefly  considered  in  this  work,
having  radii  r^  and  r,,  equal  elastic  moduli  4>,  and  for  which  the  factor
of  lateral  contraction  may  be  assumed  as  m  —  10/3,  Hertz  has  it
that:

a  The  distance  -  through  which  the  bodies  approach  one

another  normally  to  the  pressure  surface  under  the  load
P;  that  is,  the  compression  is

6  The  radius  of  the  pressure  surface,  is

o=  1.11  •! Pa   ''•'■»  [2]

c  The  maximum  pressure,  which  acts  at  the  center  of  the
pressure  surface,  is


•/  P  /n  +  r,  V


p„- 0.388  .•r.r'-^-^V  [3]


From  this  it  follows  that  the  pressure  p^  is  1.5  the  average  pressure.
7  But  the  central  elements  of  the  pressure  surfaces  are  not  only
compressed  normally  to  the  surface,  but  also  in  the  other  two  directions.
For  circular  pressure  areas  the  principal  stresses  in  the  tangential
plane  are,  according  to  Hertz,  (0.5  -f  l/m)  times  the  normal  stress,
with  l/m  =  0.3,  and  therefore  0.8  p^.
8  Hertz  carried  out  a  number  of  experiments  with  glass  in  order
to  prove  the  agreement  between  practical  results  and  his  deductions.
He  showed,  by  pressing  a  glass  lens  against  a  plane  disk  of  similar
glass,  that  the  diameters  of  the  pressure  surfaces  increased  as  the
cube  roots  of  the  pressures.  To  determine  the  influence  of  the  elastic
modulus,  he  pressed  a  steel  lens  against  the  plane  surfaces  of  various
metals,  but  encountering  difficulties  of  observation  he  gave  up  the.
experimental  proof.
422  BALL  BBABINQS

9  Later,  Auerbach  by  also  pressing  lenses  against  plane  disks
using  three  varieties  of  glass  and  one  mountain  crystal  and  measur-
ing the  compression  areas,  confirmed  the  relation  of  the  compression
area,  diameter,  load,  and  the  lens  curvature  radius  as  previously
investigated  by  Hertz.
