conspicuous effects in times which appear to us very short, 

 since in the ordinary air about us each molecule meets with a 

 million of encounters in something like the seventh part of 

 the thousandth of one second of time. In attempting to 

 interpret the results furnished by our dynamical investiga- 

 tions, the extraordinary * rapidity with which the molecular 

 events succeed one another in the actual gases of nature must 

 be fully allowed for. 



Another matter to be kept carefully in mind is that most of 

 the dynamical investigations go on the assumption that inter- 

 action between molecules, and the interaction within a mole- 

 cule 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 un- 

 ceasing setherial 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 gene- 

 ralized coordinates 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 f of these coordinates, 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 vis viva shall be 

 a sum of squares of these momentoids multiplied by functions 

 of the coordinates ; 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 

 momentoids, 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 



* See the description of an illustrative model in the last paragraph of 

 this paper. 



t A momentum may be defined as the differential coefficient of one of 

 the coordinates with respect to time, multiplied by a coefficient which 

 may be any function of the coordinates. 



