238 Mr. J. J. Thomson on the 



the primary cause. We must remember that even though 

 there were no disturbance two vortex-rings would only stay 

 together for an infinitely long time if the shortest distance 

 between their circular lines of vortex-core were infinitesimally 

 small compared with the radius of the ring ; if this were not 

 the case, two rings would separate of themselves after a time, 

 which depends on the ratio of the shortest distance between 

 their circular axes to the radius of the ring. Thus even if the 

 rings suffered no collisions they would come apart after a time; 

 and the experiments seem to show that some such cause as 

 this is, in some cases of dissociation, more effective than the 

 collisions. 



§ 1. We shall now go on to consider the subject analytically; 

 we shall take a very simple case to begin with, and treat it in 

 great detail. The case we shall begin with is that of the dis- 

 sociation of a simple gas whose molecules are diatomic, and 

 we shall suppose that the gas is a mixture of atoms and 

 molecules. The problem of finding the differential equations 

 which would be true for all stages of chemical combination 

 would lead to results of great complexity ; we shall content 

 ourselves with investigating the case when things have settled 

 down into some degree of regularity. We shall suppose that 

 the circumstances have become such that during the mean 

 time a molecule is paired the change in the number of mole- 

 cules is only a small fraction of their whole number. This 

 will not be the case when violent chemical action first com- 

 mences, but things will soon settle down into a state when 

 this will be true. We shall also suppose that all the atoms 

 remain paired for the same time t and free for the time T : 

 this of course is not, strictly speaking, the case, as the time 

 the various molecules are paired will be different; but if t is 

 proportional to the mean paired time, and T to the mean free 

 time, the general results obtained on this supposition will 

 agree with those obtained on the more accurate hypothesis, 

 which it is impossible to use, as we do not know the expres- 

 sion which tells how many molecules have any particular 

 paired time. 



To go back to the case of the dissociated gas, let m be the 

 number of molecules at any time, n the number of atoms at 

 the same time ; then if the gas is in a closed vessel, n + 2m 

 will be constant and equal to N, the number of atoms there 

 would be if all the molecules were dissociated. Let t and T 

 have the same meaning as before. T will evidently be in- 

 versely proportional to n ; let it equal r/n. Now consider 

 what happens in the time St: in this time mct/t of the mole- 

 cules will be split up ; for, as we suppose the rate of forma- 



