Sronry— Of the Kinetic Theory of Gas. 307 
tion between molecules, and the interaction within a molecule of 
one part of the molecule upon another are the only forces that 
intervene; whereas in all actual gases there is also a continuous 
interchange of energy going on between some of the internal 
events of the molecules and the ocean of unceasing etherial 
undulations in which they are immersed. 
One most remarkable and instructive dynamical theorem is 
due to the keen insight of Clerk Maxwell and of Professor 
Boltzmann. 
Maxwell discovered the important theorem that if generalised 
co-ordinates be used to represent the motion of any system of 
bodies, and if the vis viva can be expressed as a sum of squares 
of momenta’ of these co-ordinates, then the average energy will, 
if once equally divided among the terms of this series, continue 
to be so divided. 
Boltzmann has extended this theorem into the following :— 
If the vis viva can be expressed as a symmetrical function of 
the second order of the momenta (which may include both 
squares and products) then momentoids—linear functions of the 
momenta—can be so constructed that the wis viva shall be a sum 
of squares of these momentoids multiplied by functions of the 
co-ordinates: and the average energy, if at any time equally 
divided, will thenceforth continue to be equally divided between 
the terms of this expression. 
Under the Maxwell Theorem it is between the momenta, 
under the Boltzmann-Maxwell Theorem it is between the momen- 
toids, that the equal partition of energy takes place. The number 
of the momentoids is the same as of the momenta, and each of 
these latter is associated with a distinct degree of freedom in the 
system. Hence, when the Maxwell Theorem holds, the energy is 
equally divided among the degrees of freedom; but it is between 
certain quasi-groups of these degrees of freedom that the parti- 
tion takes place under the Boltzmann-Maxwell Theorem. These 
quasi-groups, like the momentoids which define them, are of the 
same number as the degrees of freedom. 
1 A momentum may be defined as the differential coefficient of one of the co-ordi- 
nates with respect to time, multiplied by a coefficient which may be any function of 
the co-ordinates. 
