TV left hand side denotes wluit we shall call the mkuu aceekrat wn
of tb- kimde energy of the partiele. The significance of the inean values will be more easilv realised if we tliink of a number of particles (listri-
f/» n cf.itinnnrv ln.W. tllft nrobabilitv ot <nven coordinates
uuicu aucuiviiiig i-w tv , , j
and veloeities being proportional to the nnmber of particles having those
coordinates tV" veloeities.
The conditions for a stationary distribution require tliat the mean acceleration of energy shall be zero just the same as conditions of equilibrium in statics require tliat the accelerations of the bodies of the system shall be zero. The equations representing these conditions we shall call equations of eucnjy equiUhrium. For the present case the equation of energy equilibrium gives
r. ,-| _ [O J
U"" J 9[*r-\
We see at once tliat energy-equilibrium is impossible unless the mean value of (P-V (l.i- is positive. But even if the partiele is moving in a region in which iPJ'idu;2 is everywhere negativo tliis condition may still be bron glit about by supposing the region bounded by perfectly elastic walls the effect of the forccs called into play dnring impact being to increase the mean value of d~ / by a tinite amount.
The condition of stability of energy equilibrium is obtained by
ü""] = ''• + '
where 'J'0 is the mean kinetic energy determined by the equation of energy equilibrium, and f is a smal 1 variation in the kinetic energ} vliicli may be due to initial disturbance. We tlius obtain
(Pe 2 r(PVl
dfi M L ftj'~ J
Tor stability the variations in e must be periodic, and tliis condition will be satisfièd if 01- Vhlx1] is positive. Thus the condition of energy equilibrium involves in tliis instance the condition of stability. II the