Probabilities to the Movement of (Jus-Molecules. 2G1 



As before, one side of the equation multiplied by a (negative) 

 constant supplies the solution *. 



That is, not taking account of the moments. When they 

 are similarly treated, there is obtained for the required 

 function one identical with (21). 



The character of the reproductive collision may be seen by 

 looking at figure 1 from the top of the page as if it were 

 the bottom. It' the velocities just after collision are reversed 

 and the figure is then looked at from above, the circle with 

 centre c which was at first overtaken now overtakes the 

 other. The dotted circle with point of impact at I' is meant 

 to represent such a collision viewed from below. The argu- 

 ment involves an assumption of the kind exhibited in the 

 simpler case (16), namely that (in a molecular medley) of 

 the couples with respective velocities U, V and u,v there are 

 (on average) as many with one molecule before as behind the 

 other. 



The arguments which have been employed with reference 

 to collisions between disks of different musses may of course 

 be narrowed to the case of perfectly similar disks, and may 

 be extended to the case of more than two types of molecules , 

 and to three dimensions. 



C. "We go on from Cartesian to Lagrangian co-ordinates, 

 beginning with the case of two dimensions and two sets of 

 molecules. 



I. Let the generalized co-ordinate of a molecule for one 

 set be Qi, Qo-.. Qm', for another q^ q 2 ...q n ; and the corre- 

 sponding components of momentum P l5 P 2 ... jo l5 p 2 



Let R be the impulsive force at the moment of collision 

 between two molecules of different type — its direction 

 being normal to the contour of each figure at the point 

 of impact, say I. Let X be the abscissa of I with respect 

 to fixed rectangular axes = 4>(Qi, Q 2 ... Q m )j and likewise 

 ¥=¥((&, Q 2 ... Q„). Let ® t and likewise V t denote the 

 respective differential coefficients of <E> and ty with respect 

 to Q*. If 6 is the angle made with the horizontal by the 

 normal at the moment of collision, Ii(<&t cos + Wt sin 0), 

 say (23) RL/=(P* — P'*) ; where, K being positive, L is 

 negative when P'>P. Like equations are obtainable for 

 the other P's. Likewise for the ]/s 11(0* cos ^-f^sin 6), 



* The' argument may be more closely assimilated to that under 

 head A by transforming- to new axes, one making an angle 6 with 

 the horizontal and the other with the vertical. Only the velocities 

 in the first of these directions, say W and w, will be changed by 

 the collision, say to W and w'. W— w=w'— W', as the argument 

 requires. Cp. C.II. below. 



