FOUNDATIONS OF THE KINETIC THEORY OF GASES. 269 



Denoting by a bar quantities referring to the critical point, these equations give 



_ = A-Re£ _ C 



Q A-Rei C 



(v + a) 3 (v + yf 

 whence 



A--Ret= P (d + a)3 , C = P(r ° +ry)S 

 a—y a— 7 



But the first equation of this section can be written as 



1T A-Re?. 



pv 



=K(i+^)(t-t) + m- J ^+-SL 



\ v + aP v + a v+<y 



+7 



By the help of the values of A — Ret, and C, just found, and the further condition that 

 p, v, t satisfy this general equation, we can easily put it in the form 



p=p h. <?-*Y > R(l + 4-Y— * (C) 



r r \ v(v + a)(v+y)/ V v + a) v v ' 



There are seven constants in this equation : — viz., p, v, t, a, y, e, and B, ; but there are 

 two relations among them, one furnished by the usual condition that the gas treated has 

 unit volume at 0° C, and 1 atm. ; the other (from the conditions of the minimax) being 



3v + a + y = — - 

 P 



73. If we compare (C) with the corresponding forms of the equations of Van der 

 Waals and Clausius ((A) and (B) of § 66 above) we see that all three agree in a remark- 

 able manner as to the form of the equation of the critical isothermal. In fact, the only 

 difference is that in (C) the divisor of (v — v) z contains three distinct factors, while in 

 each of (A) and (B) two of the three factors are equal. It is quite otherwise with the 

 term which expresses the difference of ordinates between the critical isothermal and any 

 other of the series : — so that even if all three equations agreed in giving the correct form 

 of the critical isothermal no two of them could agree for any other. 



XXIII. — Comparison with Experiment. 



74. We must now compare our formula with experiment. And here I have been 

 exceptionally fortunate, as the kindness of M. Amagat has not only provided me with a 

 complete set of values of pv in terms of p for C0 2 between the limits 1 to 1000 atm. and 

 0° to 100° C, but has further replied to my request for a set of values of p, at different 

 temperatures, for certain special values of v. This important table I give in full, inserting 

 columns of differences. It is very much better adapted than the former to numerical 

 calculation, as the form of the virial equation requires that v should, for this purpose, 

 be treated as the independent variable. 



