by  Forces  of  Relatively  Long  Duration.  291 
and  the  ratio  (R)  o£  this  to  the  energy  before  collision  is 
'        21      Ly2  ap  3    Wa0  .qnv 
64  a2  n°  pr6       32  (1  —  o-)"  a2  nP  prl 
if  we  introduce  the  value  of  k2  from  (2). 
The  precise  value  of  a2  would  have  to  be  calculated  from 
Lamb's  theory.  It  is  easy  to  see  that  it  is  decidedly  smaller 
than,  but  of  the  same  order  of  magnitude  as,  the  mass  of  the 
sphere,  viz.  ^irpr3. 
The  precise  value  of  R  would  depend  also  upon  a,  but  for 
our  purpose  it  will  suffice  to  make  a  =  J.     Thus  we  take 
R  =  S-?V (31) 
op*  n°  r3 
According  to  Lamb's  calculation  *  for  the  principal  vibra- 
tion of  the  second  order  in  spherical  harmonics  (a  =  -J) 
■VC!)^-  ■  ■  ■  •  <K> 
so  that  approximately 
E  =  50  7WJ) (33) 
In  (33)  */  (E  /  p)  is  the  velocity  of  longitudinal  vibrations 
along  a  bar  of  the  material  in  question,  and  the  comparison 
is  between  this  velocity  and  the  velocity  of  approach  before 
collision.  In  steel  the  velocity  of  longitudinal  vibrations  is 
about  500,000  cms.  per  second,  or  about  16  times  that  of 
sound  in  air.  It  will  be  seen  that  in  most  cases  of  collision 
R  is  an  exceedingly  small  ratio. 
The  general  result  of  our  calculation  is  to  show  that 
Hertz' s  theory  of  collisions  has  a  wider  application  than 
might  have  been  supposed,  and  that  under  ordinary  conditions 
vibrations  should  not  be  generated  in  appreciable  degree. 
So  far  as  this  conclusion  holds,  the  energy  of  colliding 
spheres  remains  translational,  and  the  velocities  after  impact 
are  governed  by  Newton's  laws,  as  deducible  from  the 
principles  of  energy  and  momentum. 
Terling  Place,  Witliam, 
Dec.  22,  1905. 
Loc.  cit.  p.  206. 
U2 
