328 Aggregation and Dissociation [CH. xvni 



and write 



u 2 + v 2 + w 2 = c 2 , of + /3 2 + 7 2 = V 2 , 



we can transform the foregoing expression into 



A 2 e- 2hmC<2 dudvd\ffda;dydze-l hmV2 -' 2h *dctdfidv4!Trr 2 dr (771). 



The first factor after the A 2 expresses the law of distribution of transla- 

 tional velocities for a double molecule. It is exactly the same as if each double 

 molecule were a permanent structure of mass 2m. The remaining factors 

 express the distribution of those coordinates which may be regarded as 

 internal to the double molecule. 



395. Throughout the motion of a double molecule, so long as it is undis- 

 turbed by collisions, c 2 will remain constant, so that from the energy equation 

 it follows that raF 2 + 2SP remains constant. The possible orbits which the 

 component molecules can describe about their common centre of gravity 

 fall into two classes, according as they pass to infinity or not. Analytically 

 these two classes are differentiated by the sign of |raF 2 + 2" v I r . Double 

 molecules for which ^wF 2 + 2 x P is positive consist of two molecules which 

 have approached one another from outside each other's sphere of action, and 

 which after passing once within a certain minimum distance of each other, 

 will again recede out of each other's sphere of influence. On the other hand, 

 double molecules, for which JmF 2 4- 2^F is negative, consist of two molecules 

 describing orbits about one another, these orbits being entirely within the 

 two spheres of action, and this motion continues except in so far as it is 

 interrupted by collisions with other molecules. It is clear that double mole- 

 cules of the first kind are simply pairs of molecules in collision. In discussing 

 molecular aggregation we must confine our attention to double molecules of 

 the second kind, i.e. those for which |raF 2 + 2"^ is negative. It is to be 

 noticed that double molecules of this kind cannot be produced solely by the 

 meeting of two single molecules. It is necessary that while the single mole- 

 cules are in collision something should happen to change the motion in fact 

 to change the sign of ^mF 2 + 2^. This might be effected by collision with 

 a third molecule, or possibly if ^mF 2 + 2^ were very small at the beginning 

 of an encounter, sufficient energy might be dissipated into the ether for 

 ^mF 2 -f 2^ f to become negative before the termination of the encounter. 

 We shall return to the consideration of this second possibility later. If this 

 were the primary cause of aggregation, we should no longer be able to use 

 the equations with which we have been working, since they rest upon the 

 assumption of conservation of energy. 



396. Integrating expression (770) over all values of u, v and w, we find 

 for Vi, the molecular density of uncombined molecules, 



