276 Free Path Phenomena [CH. xv 



this being the only way of combining in, C and p, so as to get a quantity 

 of the same physical dimensions as K. If K is observed to vary as the 

 rcth power of the absolute temperature, and therefore as the 2nth power of 

 C, we have the relation 



fT\ " ~I " / /* A 1 \ 



2w = (641), 



S *~~ J. 



the same relation as is given by equation (584). 



From experimental evidence we found values for s ranging from 5'2 to 12. 

 Maxwell* believed the value of n for air and some other gases to be n= 1, and 

 hence supposed that s = 5 gave the true law of force for all gases. It is now 

 known that this is not true, but nevertheless the law s = 5 gives an interesting 

 mechanical illustration of the theory of gases, and is moreover the only law 

 for which the theory has been worked out with strict mathematical accuracy. 



If we make the assumption of a repulsion varying as some inverse power 

 of the distance, the value of s for all the gases tabulated on p. 257, lies 

 between s = 5 and s = oo . Now the law of force for elastic spheres is one in 

 which the force is zero for r > a, and is infinite for r <&. This may suitably 



/<r\ s 

 be represented by the function /* (-] where s = co . Thus s = oo (the law 



worked out in previous chapters) and s = 5 (the law worked out in the 

 present chapter), may be regarded as limits between which all gases lie. 



GENERAL EQUATION OF TRANSFER. 



330. Let Q be any function of the velocity components of a single 

 molecule, e.g. momentum, energy. We proceed to form general equations 

 expressing the transfer of Q. 



At any point x, y, z let Q be the mean value of Q, so that 



(u, v, w)Qdudvdw (642). 



If we consider a fixed rectangular parallelepiped dxdydz at this point, 

 the number of molecules inside it is vdxdydz, and hence 2Q, the aggregate 

 amount of Q inside the element of volume, is given by 



We now examine the various causes of change in 2Q. In the first place 

 some molecules will leave the element dxdydz, taking of course a certain 

 amount of Q with them. It has been already found in expression (348), that 

 the total number of molecules of class A lost to the element dxdydz in time 

 dt is 



dxdydzdt u^-+v^--\-w^- (vf)dudvdw, 

 * " On the Viscosity of Internal Friction of Air and other Gases," Collected Works, n. p. 1. 



* 



