ON THE KINETIC THEORY OF GASES. 217 



approaching- each other reach the distance a apart (in which 

 a is, of course, greater than s, the diameter of a molecule) 

 an impulse acts in the line of centres increasing the square 

 of the relative velocity u by the constant quantity c 2 , sup- 

 posed to be independent both of u and of the angle which 

 its direction makes with the line of centres at the instant. 

 When the distance is less than a no force acts. This ap- 

 proach is called an encounter. If the value of u, and the 

 angle between its direction and the line of centres, be suit- 

 ably chosen, there will be also an impact between the hard 

 nuclei. Whether an impact takes place or not, the mole- 

 cules will again on separation reach the distance a. And 

 then the square of the relative velocity is again diminished 



bv c 2 . Also c 2 =—r,e. e beiny one of the constants in 

 3 Jk s 



Tait's virial equation above given. 



28. On this hypothesis, if c 2 , a and s were all known, 

 the motion of the system for any assigned values of t and v 

 would be determinate, whether our mathematical powers suf- 

 fice to make all the necessary calculations or not. We should 

 then have only three disposable constants corresponding to 

 c 2 a and s to introduce into the virial equation. So 

 comparing Tait's hypothesis with the practical formula 

 which he derives from it, we seem to have in the latter 

 two constants too many. To adapt the hypothesis to 

 the virial equation, we have to find (1) the modification 

 of the term l,mu 2 , (2) that of 2^Rr. Now l^mir, or the 

 mean kinetic energy, is increased, approximately at least, 

 by adding c 2 for the number of pairs which at any (and 

 therefore every) instant are at less than distance " a " from 

 each other, are in "entanglement" as he defines it. The 

 value of 1mu 2 depends then on the average duration of an 

 entanglement, which he calculates, part iii., p. 1037. The 

 term 2^Rr has to be modified in two ways. Firstly, calcu- 

 late the average value of the attractive impulse for each 

 encounter, and multiply by the distance ''a'' at which it 

 takes place. That calculation is performed, pages 1033, 

 1034. The result will be of the opposite sign to the term 

 ?mr. Secondly, remembering that for some encounters 



V 



