for a System of Correlated Variables. 23 



If the gas be rare enough, the last or Virial term is negligible, 

 because R is negligible for all but an infinitely small proportion 

 of the pairs of molecules at every instant. For such a gas then 



very approximately. That is, Boyle's law is fulfilled very 

 approximately. This is the case for air and other gases at 

 ordinary pressures. For such gases, since R is, for nearly all 

 pairs, negligible, the correlation coefficients b are, for nearly 

 all pairs, negligible, and Maxwell's law holds. But as the 

 density of the gas increases, the resistance to compression 

 increases beyond — and ultimately very far beyond — what it 

 would be under Boyle's law (see Lord Kelvin's paper " On 

 the Problem of a Spherical Gaseous Nebula," p. 2 GO note). 

 For a sufficiently dense gas therefore the Virial term can no 

 longer be neglected, as a considerable part of the pressure P, 

 in §Pv, is due to it. For the same reason the correlation 

 coefficients />, which are proportional to the R's, can no longer 

 be neglected. Q becomes a complete quadratic function con- 

 taining products as well as squares of the velocities. 



40. Experimentally, by reason of the homogeneity, any 

 portion of the gas, if containing a sufficiently great number 

 of molecules, has the same properties as any other portion, or 

 as the whole. Analytically the same result appears thus : 

 About any point in the gas as centre, suppose a sphere of 

 radius c containing N molecules, and another concentric 

 sphere of radius c + c', containing X + N' molecules, and both 

 c and c much greater than the radius of correlation. Let 

 Q N be the value of Q. for the N molecules in the c sphere, 

 Qn+n' its value for the N + N' molecules within the 

 sphere c + c'. Then, if N be very great, 



Qx_ N_ 



Q>- + v """N+N" 



Therefore ^rp^, , or A, is independent of N. It is true the 



number of roots of equation B, as applied to the N molecules, 

 is N, and as applied to the N + N' molecules is N + N', or N' 

 new roots are introduced by taking in the additional N' 

 molecules. The statement that \ is independent of N means 

 therefore that all the roots of (B) , as applied to the N molecules, 

 are also roots of (B) as applied to the N + N' molecules. And 

 in particular the least root of (B) is the same in the two cases. 

 The correlation coefficients between the N' new variables and 

 the original N are, except in an infinitely small proportion of 

 the cases, evanescent. Therefore by art. 35 the normal state 



