FOUNDATIONS OF THE KINETIC THEORY OF GASES. 89 



compression than that Law indicates. But Andrews showed that the same 

 thing holds for all gases at temperatures and pressures over those correspond- 

 ing to their critical points. And Amagat has experimentally proved that in 

 gaseous hydrogen, which has not as yet been found to exhibit any traces of 

 molecular attraction between its particles, the graphic representation of pY 

 in terms of p (at least for pressures above an atmosphere, and for common 

 temperatures) consists of a series of parallel straight lines. If this can be 

 accounted for, without the assumption of molecular repulsion but simply by 

 the impacts of the particles, a real difficulty will be overcome. And it is 

 certain that, at least in dealing with hard colliding spheres if not in all cases, 

 we have no right to extract from the virial, as the pressure term, that part 

 only which depends upon impacts on the containing vessel ; while leaving 

 unextracted the part depending on the mutual impacts of the particles. The 

 investigation which follows shows (so far as its assumptions remain valid when 

 the particles are not widely scattered) that no pressure, however great, can 

 bring a group of colliding spheres to a volume less than four times the sum of 

 their volumes. If they were motionless they could be packed into a space 

 exceeding the sum of their volumes in the ratio 6 : rr^/2, or about 1-35 : 1, 

 only.] 



In the case of hard spheres we have obviously r = s; and, with the notation 

 of § 19, remembering that Q = P, k = h, we have 



E=-P(u-v). 



Hence we must find, by the method of that section, the mean value of the 

 latter expression. It is easily seen to be 



_ -afwiV* sin /3 dfi cos 2 >y sin yd<yd<j> _ 2V/vv 1 v HvJvv 1 

 fvv x v^ sin ft d/3 cos 7 sin <yd<yd<f> 3 fw^o^-dv^vv^ 



2P I 4 /4 



3 I 3 /3 ~ l V 2h' 



But, § 14, the average number of collisions, per particle per second, is 



2 N 



2 f± ^ 



Hence, for any one particle, the sum of the values of E (distributed, on the 

 average, uniformly over its surface) is, in one second, 



v , m 2NP , 4N _ 



2(R) = — ^v ~T3 v IWs2= -P-4**' 



Thus it would appear that we may regard each particle as being subjected to 

 the general pressure of the system ; but as having its own diameter doubled. 



VOL. XXXIII. PART I. M 



