166 Prof. G. H. Bryan on certain Dynamical 
exists requires that the mean value of the term ud°"Y/dt dx 
should also vanish, reducing the energy-acceleration equation 
in this case to the ordinary standard form, brackets denoting 
mean values : — 
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For energy equilibrium the left-hand side must vanish. 
This is only possible when the mean value of d^Y/dx 2 is 
positive. Now the assumed absence of directional properties, 
or in other words the assumed isotropic character of the 
medium, requires that the average values of d 2 Y/dx 2 , d 2 Y/dy 2 , 
and d 2 V/dz 2 shall be equal ; therefore each is equal to one- 
third the average value of AV. We conclude that statistical 
energy equilibrium cannot exist in a system of particles 
possessing the assumed properties unless AV is positive. 
For the Newtonian law of force AV = 0. In this case 
the mean value [d 2 Y/dx 2 ] therefore also vanishes, and the 
energy-acceleration equation takes the form : — 
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This shows that the only kind of statistical equilibrium is 
the statical state of unstable equilibrium defined hydY/dx = 0, 
dYjdy = 0, and dV/dz — on each particle, and in the 
absence of this state an acceleration of kinetic energy will 
take place. 
It is thus impossible with the Newtonian law to build up 
a medium of material particles, either attracting or repelling 
one another, and possessing properties of energy equilibrium 
analogous to those of a system of gas molecules. 
It need not be pointed out that this investigation does not 
preclude the possibility under the Newtonian Law, of 
stationary motions such as those occurring in the solar 
system. 
Exactly the same conclusions hold good if the law of force 
is such as to make AY negative. The equations of energy 
equilibrium here require the further condition that [u 2 ~\ [v 8 ] 
\_w 2 ~\ should be zero, which can only happen if the system 
always remains at rest. 
The necessary condition AV>0 becomes, for the particular 
case in which the field is due to an attracting particle placed 
at the origin, 
r 2 dr\ dr 
showing that r 2 dY /dr must increase with r. In other words, 
the force if repulsive must vary according to a higher power 
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