Partition of Kinetic Energy. 227 



kinetic energy. For it is easy to construct a system in which 

 that is not true. For instance, suppose two parallel fixed 

 elastic planes, P and Q, and PQ a common normal. Let m 

 and M be two elastic spheres of unequal masses, each having 

 its centre in the line PQ. Initially let m be touching the 

 plane P and be projected in direction PQ. Simultaneously 

 let M be touching the plane Q and be projected in direc- 

 tion QP. The two spheres will collide, having their line of 

 centres in PQ, with the point of contact, say, at R. If the 

 momenta are equal and opposite, the two spheres will, after 

 collision, retrace their respective courses, will rebound nor- 

 mally from the two planes, and will again collide at the same 

 place, and so on for ever. Since the momenta are equal and 

 the masses unequal, the kinetic energies are unequal. Yet 

 the motion is stationary. The condition of stationary motion 

 is then not a sufficient foundation for the theorem of equal 

 partition of energy. What further condition is necessary or 

 sufficient ? 



It is generally understood that the theorem is proved in the 

 Kinetic Theory of Gases, the simplest form of which may be 

 stated thus : — 



A very great number of elastic spheres are moving in a 

 bounded space, and freely by their encounters or collisions 

 exchanging energy with one another. Let no forces act 

 except during collisions between two spheres, or between a 

 sphere and the bounding surface, supposed perfectly elastic. 

 Let /(«, V) w)dadv dw or / . dadv dw be at any instant the 

 number of spheres m whose component velocities lie within 

 the limits 



u . . . u + du &c (1) 



Similarly, let F(\J,V,W) d\J dV dW or F.dJJdVdW be the 

 number of spheres M whose component velocities lie at the 

 same instant within the limits 



U...U + dU&c (2) 



If m and M, having these velocities, are properly situated 

 with respect to one another, they will within the time dt after 

 the given instant collide with each other, and assume new 

 velocities denoted by the accented letters it' &c. U 7 &c. where 



m (u 2 + v* + iv 2 ) + M ( U* + V 2 + W a ) = m {vl 2 + v H + w'*) 



+ M(U'* + V /2 + W' 2 ). 



Boltzmann now says that it is necessary to make a special 

 assumption. And the assumption he does make is that the 

 motion is, and continues to be, molecular unyeordnet (see 

 Vorlesungen ilber Gas i heorie, Part I.). The state " molecular 

 ungeordnet" is not completely denned, perhaps because if it 



Q2 



