163-167] The Virial 147 



jyi 



If we write Nv for the modified value of n -, so that v is a modified 



Ii> 



molecular density, the corrected form of expression (337) becomes 



, r , -2/if4>(r)dr , 



%7rNvr z e ** an 



This is the number of pairs of molecules each of which gives a contri- 

 bution r</> (r) to 22r</> (r). Hence, multiplying the expression just obtained 

 by r(f> (r) and integrating over all values of r, we obtain 



r) = ZirN f i/Vty (r) *"<?/' * <r) dr dY ....... . . .(339). 



J o 



165. Progress is for many reasons impossible in the general case. Let 

 us fall back on the simplifying assumption of the existence of spheres of 

 molecular action satisfying the conditions mentioned in 82. If each sphere 

 is of diameter or, then in every term which supplies an appreciable Ncontribu- 

 tioii to 2S</> (r), the value of r is very nearly equal to <7. If we write 



<f) (r) dr = x, 



so that (f> (r) dr = dx, 



then equation (339) becomes 



r 



22r<f> (r) = - 27rN v^e'^ dx. 

 Jo 



In the integral on the right-hand side, v'r 3 may be replaced by v'a 3 and 

 taken outside the sign of integration. The value of the integral is accord- 



/ q 

 V (T 



ingly -f- and the equation becomes 



,e>A{\\ 

 (340). 



166. We next turn to the calculation of v. In virtue of the supposed 

 existence of a sphere of molecular action, we can imagine each of N 1 

 molecules surrounded by spheres of radius cr, and then every position outside 

 these spheres may be supposed equally probable as a possible position for 

 the centre of the .ZVth molecule, while any position inside these spheres is 

 impossible. Thus if fl' be the total volume excluded by these spheres, we 

 shall have 



N 

 instead of v = -^ . Obviously if we neglect the distinction between N 1 



and N, 



fl' = |AW .............................. (342). 



167. Equation (335) now becomes 



102 



