434 Ml;. S. CHAPMAN OX THE KINETIC THEORY OF A GAS 



The other hypothesis referred to seems to be much more in agn>nnrnt with fact, 

 but its consequences have been worked out less accurately. The method which 

 has almost always been used is the one originally devised by CLAVSIUS and 

 MAXWELL ; MAXWELL abandoned it later, however, as it had " led him at times into 

 grave error." In spite of its apparent simplicity, numerical errors of large amount 

 may undoubtedly creep in in a very subtle way. Hence the theory of a gas whose 

 molecules are elastic spheres remains in a rather unsatisfactory state. As a 

 " descriptive " theory (to use MEYER'S apt term) it has, however, served a useful 

 purpose ; the general laws of gaseous phenomena have been developed by its aid in 

 an elementary way, which has conduced to a wider diffusion of knowledge of the 

 kinetic theory than would have been possible if the sole line of development had been 

 by the more mathematical and accurate methods used by MAXWELL and BOLTZMANN. 



In this paper I have applied the latter methods, with an extension of the analysis, 

 to the elastic-sphere theory among others. In fact, I have obtained expressions for 

 the viscosity, diffusivity, and conductivity of a gas without assuming any properties 

 of the molecules save that they are spherically symmetrical. Many known laws are 

 thus proved more generally than in any former theories, but the formulae so obtained 

 cannot in all cases be put into a really useful form without a knowledge of the nature 

 of the molecules. The supplementary calculations required to complete the general 

 formulas of Part I. of this paper are carried out in Part II. for three special cases, 

 viz., rigid elastic spherical molecules, molecules which are centres of repulsive or 

 attractive force varying inversely as the n Ul power of the distance, and rigid elastic 

 spherical molecules surrounded by fields of attractive force. In Part III. the general 

 formulae are completed in these cases and discussed in their relations to the results of 

 former theories and of experiment. 



PART I. GENERAL THEORY. 

 1. Statement of the Problem. 



We shall deal with a gas composed of two kinds of molecules, which are all 

 supposed to be spherically symmetrical ; m, m', v, v will denote their masses and the 

 number of each kind per unit volume respectively. Similarly the velocity components 

 of typical molecules of the two kinds will be denoted by (, v, w], (', v', w'}. Let Q 

 be any function of the velocity components of a single molecule (e.g., momentum, 

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



Q = I 1 1 Qf(u, v, w) du dv dw, 

 where f(u, v, w) is the function which expresses the law of distribution of the 



