32-36] The Method of General Dynamics 35 



The Motion in the Generalised Space. 



35. The general nature of these paths can be seen without trouble. 

 A collision either of two molecules, or of a molecule and the boundary, occurs 

 when, and only when, a path meets one of the surfaces of the regions 

 excluded in 33. Now between collisions every molecule moves with uniform 

 velocity in a straight line. Thus if at time t = 0, the coordinates of a system 

 are 



a'i y*> Za, Ua, V a ', W a ', OC b ' , y b ', Z b ', U b , V b ', W b ............... (54), 



the coordinates at time t, assuming that no collision has taken place in the 

 interval, will be given by 



a = Xa' + u>a't, y a = y a ' + v a 't, etc. 

 u a = u a r , v a = v a ', etc. 



To find the equations of the path described by the representative point in 

 the generalised space we eliminate t, and so obtain 



/ / - ...... t /K-\ 



Ua Va Y .................. (55), 



Ua = U a ', V a = V a ', W a = W a ' ... etc. ' 



and since these equations are linear they are of course the equations of a 

 straight line. We therefore see that the paths in the generalised space are 

 rectilinear except when they meet the excluded regions. Along the recti- 

 linear parts of any paths, all the coordinates u a , v a , w a , u^ ... etc., maintain '- 

 constant values, and any series of paths for which these constant values are ,, 

 the same are all parallel. When a representative point, moving along one 

 of these paths, meets a boundary of the excluded space corresponding to a 

 collision it must be supposed to move along this boundary until it reaches 

 the point of which the coordinates are those of the system after collision, and 

 then to start from here and describe the new rectilinear path through this 



point. 



/ 



36. Now in the gas of the Kinetic Theory, we do not know anything as 

 to the coordinates of the individual molecules of the gas : the problem we 

 have to attack is virtually that of finding as much as we can about the 

 behaviour of a dynamical system, without knowing on which of the paths in 

 our generalised space its representative point is moving. 



Our method is therefore to start an infinite number of systems, each 

 system being a complete gas of the kind already specified, so as to have 

 systems starting from every conceivable configuration, and moving over 

 every path ; and to investigate, as far as possible, the motion of this series of 

 systems, in the hope of finding features common to all. Or, what comes 

 to exactly the same thing, we shall imagine our generalised space filled 



