470  Notices  respecting  New  Books. 
approximation  to  their  real  conditions  is  attempted.  In  all  actual 
cases,  however,  surfaces  are  capable  of  exerting  a  tangential  as  well 
as  a  normal  reaction ;  and  the  conditions  of  these  cases  are  much 
more  closely  represented  if  it  is  assumed  that,  as  well  as  a  normal 
reaction  (R),  there  is  also  a  friction  or  tangential  resistance  (F) 
which  acts  in  any  direction  needed  to  oppose  sliding,  and  of  any 
amount  needed  for  equilibrium  up  to  a  limiting  value  fiR,  where  fi, 
the  coefficient  of  friction,  has  a  definite  numerical  value  depending 
on  the  surfaces  of  contact.  The  object  of  Mr.  Jellett's  Treatise  is 
to  discuss  the  properties  and  effects  of  this  force  as  a  part  of  rational 
mechanics,  and  not  merely  to  consider  it  as  a  force  by  reason  of 
which  certain  corrections  have  to  be  applied  to  results  obtained  on 
the  supposition  of  perfect  smoothness  before  those  results  are  capable 
of  useful  application  (p.  v), — in  other  words,  to  trace  out  the  conse- 
quences that  follow  from  the  assumption  that  the  reactions  follow 
this  law  absolutely. 
The  contrast  between  the  usual  and  the  more  exact  assumptions 
can  be  made  as  follows  : — Suppose  a  point  acted  on  by  any  forces  to 
rest  against  a  plane  ;  let  Q  be  the  resultant  of  all  the  forces,  the  re- 
action of  the  plane  excepted ;  let  a  cone  be  described  round  the  nor- 
mal as  axis,  with  the  point  as  vertex,  and  semi-  vertical  angle  equal  to 
the  angle  of  friction  (tan-1  /*).  If  the  plane  were  smooth,  the  con- 
dition of  equilibrium  would  be  that  Q  act  along  the  normal ;  but  if 
the  plane  is  rough,  Q  will  be  balanced  by  the  reaction  of  the  plane, 
provided  it  act  along  any  line  within  the  cone.  If  Q  act  along  a 
line  on  the  surface  of  the  cone,  the  point  is  in  an  extreme  position, 
i.  e.  it  is  on  the  point  of  sliding*.  This  does  not  seem  a  very  serious 
modification  of  the  conditions;  but  when  its  consequences  are 
worked  out  in  any  particular  case  they  frequently  show  the  problem 
in  an  entirely  new  light.  Speaking  generally,  the  effect  is  to  in- 
troduce into  the  solution  quantities  which  are  indeterminate  and 
limited  by  one  or  more  inequalities,  instead  of  quantities  which  would 
admit  of  exact  determination  if  the  surfaces  were  smooth.  The 
solution,  however,  ordinarily  becomes  determinate  if  we  suppose  it 
to  be  made  with  reference  to  an  extreme  position ;  but  even  then 
the  solution  is  generally  quite  different  from  what  would  be  obtained 
on  the  supposition  of  smoothness.  Accordingly,  in  chap.  ii.  and  iii., 
Mr.  Jellett  treats  the  subject  under  two  heads  :  he  discusses  (1)  the 
conditions  to  be  fulfilled  when  equilibrium  exists,  and  (2)  the  condi- 
tions under  which  bodies  are  in  an  extreme  position. 
Passing  from  the  case  of  equilibrium  to  that  of  motion,  the  fol- 
lowing points  are  discussed  in  chap.  iv.  and  v. : — the  motion  of  a 
particle  and  system  of  particles  on  rough  lines  and  surfaces,  that  of 
a  solid  body  on  a  rough  plane,  and  the  initial  motion  of  a  system  of 
particles,   and  of  a  solid  body  on  rough  surfaces.     Indeterminate- 
*  This  geometrical  conception  is  due  to  the  late  Canon  Moseley,  an 
author  who  many  years  ago  strongly  insisted  on  the  need  of  taking  account 
of  friction  in  mechanical  questions,  and  exemplified  his  views  in  a  very 
elaborate  discussion  of  the  Theory  of  Machines  which  forms  part  of  his 
'  Mechanical  Principles  of  Engineering.' 
