Xotices  respecting  New  Books.  471 
ness  meets  us  in  questions  of  motion  as  well  as  in  questions  of  equi- 
librium, e.  g.  in  the  case  of  the  initial  motion  of  a  system  of  parti- 
cles, Mr.  Jellett  points  out  that,  "  it  is  easily  shown  by  the  principles 
of  ordinary  dynamics  that,  if  the  supporting  surfaces  be  smooth,  the 
initial  motion  and  all  subsequent  motions  are  perfectly  determi- 
nate;" and  when  the  surfaces  are  rough,  "the  equations  of  the 
problem  enable  us  to  determine  completely  the  condition  of  the 
moving  particles,  if  we  know  that  they  are  moving.  But  as  these 
equations  do  not  give  us  any  means  of  deciding  which  particles  are 
at  rest  and  which  are  in  motion,  the  initial  motion  of  the  system .... 
remains  still  indeterminate."  The  amount  of  this  indeterminateness 
can  be  reduced  by  rejecting  certain  systems  of  movements  ;  but  when 
all  "have  been  rejected  except  such  as  are  both  geometrically  and 
dynamically  possible,  it  will  be  frequently  found  that  the  question 
remains  still  indeterminate  ;  e.  g.  this  will  be  in  general  the  case 
when  there  exists  an  equation  of  condition  involving  only  the  coor- 
dinates of  quiescent  particles  "  (pp.  104,  105,  109). 
The  sixth  chapter  is  devoted  to  the  question  of  necessary  and  ^os- 
sible  equilibrium — a  distinction  to  which,  we  believe,  Mr.  Jellett  has 
been  the  first  to  draw  attention  ;  what  is  meant  by  it  will  be  best 
understood  by  considering  a  particular  case : — Suppose  0  to  be  a 
point  in  front  of  a  vertical  wall ;  a  weightless  rod  is  fixed  by  one  end 
to  O  and  can  turn  freely  round  it ;  it  carries  at  the  other  end  a 
heavy  point  P,  which  is  placed  against  the  wall.  If  the  rod  were 
allowed  to  turn,  the  locus  of  P  on. the  wall  would  be  a  circle;  we 
will  suppose  C  to  be  the  centre  of  this  circle,  and  A  CB  to  be  a  ver- 
tical diameter,  the  point  A  at  the  top ;  suppose  P  placed  in  some 
position  near  A,  and  let  6  and  /3  denote  the  angles  P  C  A  and  P  O  C 
respectively,  and  Q  the  reaction  transmitted  along  the  rod  from  O 
to  P.  Now  if  we  suppose  Qx  to  denote  an  angle  such  that 
tan  01=fx  cotan  ft,  then  if  U<  6}  and  a  small  motion  is  communicated 
to  P  in  this  position,  the  forces  will  tend  to  destroy  this  motion,  and 
the  position  is  one  of  necessary  equilibrium.  Next  let  the  question 
be  regarded  thus  : — By  resolution  of  forces  it  is  shown  that  the  nor- 
mal pressure  on  the  wall  is  Q  cos  ft,  and  that  the  square  of  the  re- 
solved pressure  on  the  plane  is 
g-  -  2gQ,  sin  ,3  cos  d  +  Q2  sin2  /3  ; 
so  that  there  will  be  equilibrium,  provided 
g-  —  2gQ  sin  ft  cos  6  +  Q"  sin2  /3  =  or  <  /x2Q:  cos2  ft. 
This  condition  cannot  be  fulfilled  for  any  value  of  Q  whatsoever  if 
sin  0>/i  cotan /3  :  there  is  therefore  a  limiting  value  d2  such  that 
sin  02  =  fj.  cotan  ft  ;  and  if  fi<0 '.,,  the  above  condition  will  be  fulfilled 
provided  Q  have  the  proper  value  ;  but  as  Q  is  quite  indeterminate, 
such  a  position  is  one  of  possible  equilibrium.  We  arrive,  therefore, 
at  the  following  result,  that  when  6<6V  there  is  necessarily  equili- 
brium, and  the  equilibrium  is  stable  in  the  sense  that  it  will  not  cease 
if  a  small  velocity  be  communicated  to  P  ;  but  if  6  >  6X  and  <  B .,,  there 
will  be  equilibrium  or  not,  according  to  the  actual  value  of  Q,  and  this 
equilibrium,  if  it  exist,  will  be  unstable. 
