Analogues of Temperature Equilibrium. 767 
o£ the inverse distance than the square (the inverse fifth 
power would represent a possible law of force if repulsive). 
If the force is attractive it must vary according to a lower 
power of the inverse distance than the square. 
The second point to be noticed is that unless the expression 
for the kinetic energy of a dynamical system is a quadratic 
fuuction of the velocities with constant coefficients, the 
equations of energy equilibrium no longer assume the form 
of linear relations connecting the mean squares and products 
of velocities, but they also involve terms of the fourth degree 
in these velocities. An illustration of this fact is afforded by 
the motion of a particle in a plane when referred to polar 
coordinates. If we write down the expressions for the energy- 
accelerations 
Jo*-) -a %ivh 
in terms of the velocities and coordinates, using the equations 
of motion 
•• M dV 1 d 9 • dV 
r — rd 2 = — -7- and --=- (r 2 6) = ja 
dr r at ' rat) 
we shall obtain expressions involving the velocity components 
up to the fourth degree, and the energy components there- 
fore up to the second degree. The conditions of energy 
equilibrium between the transversal and radial components 
will therefore no longer take the form of linear relations con- 
nectmg the mean values of the energy components in question. 
On the other hand, when we refer the motion to x and y 
coordinates, the equations of energy equilibrium are linear in 
the energy components. 
It appears, therefore, that if dynamical systems are to 
represent the phenomena of temperature-equilibrium consis- 
tently with the commonly accepted hypothesis that tempe- 
rature is a quantity of the nature and dimensions of molecular 
kinetic energy, the expression for the kinetic energy must 
in general have constant coefficients or must at least satisfy 
certain conditions which are fulfilled in the case of constant 
coefficients. While the analogy of kinetic energy with tem- 
perature may hold good in the case of a system of particles, 
there must exist dynamical systems for which this analogy does 
not hold good. As a purely negative conclusion this result 
is not inconsistent with Stefan's law in which we have the 
energy of radiation proportional to the fourth power of the 
energy of molecular motion. But whether it is possible to con- 
struct a dynamical model whose energy-partition is analogous 
to Stefan's Law must be a question for future investigation. 
