Mr Pocklington, On the Kinetic Theory of Matter. 289 



ables are as, y, 6, x, y. The first two terms vanish. The third is 

 found from (1) to be (x cos 6 + y sin 6) j a, while from (2) 



dx k 2 

 and from (3) and (1) 



K7 = sin v ^-r , 



ox a ox 



k 2 



(x sin 26 — y cos 26) sin 6. 



a (If + a 2 ) 

 Similarly 



I - - si&o (2> sin 2 " + * eos ^ oos '■ 



and hence 



1 C?/ a , . „ . ■ n\ 



t-t= t> o i x cos 6 + y sin 6) 



J dt ¥ + a 2 v * ' 



= e_ 



~0 

 by (3), giving J=e/e 0/ 



In the final state vanishes, and hence / is then zero. 



This result can be generalised. For, representing an infinite 

 number of systems by points in a m-dimensional space, if the 

 systems tend to a final state, these points must ultimately collect 

 on some surface or other locus of fewer dimensions than m. Their 

 density there will be infinitely greater than it originally was, and 

 J, which is proportional to the reciprocal of the density of dis- 

 tribution if the original distribution was uniform, will vanish 

 on this locus, and therefore the final state must satisfy the 

 equation J = 0. 



The converse proposition, that if / is capable of vanishing, the 

 system tends to a final state, given by J=0, does not follow, and 

 is possibly not always true. If J cannot vanish there is no 

 tending to a final state. Using the last theorem, we can readily 

 prove that there is no tending to a final state in the case of a 

 circular disc rolling on a rough surface ; the tendency found by 

 experiment for it ultimately either to roll in a straight line or 

 to spin with its plane vertical must be due to rolling friction 

 or some other cause that is neglected in forming the ordinary 

 equations of motion. 



9. In the case considered in §§ 5, 6 the final state was given 

 by 6 = 0, an equation involving a velocity. It is possible for the 

 final state to be given by an equation involving only coordinates. 

 Let us consider the case of a tricycle. We suppose that the 

 wheels and framework are massless, that a mechanical system, 

 symmetrical about the steering axis, is mounted on it, and that 

 the steering axis is connected to the framework by massless springs 



