440 MR. S. CHAPMAN ON THE KINETIC THEORY OF A <i.\S 



ft. 



Integration vnth Respect to p. 



The integrand of equation (8) contains p explicitly in the term p, and implicitly 

 in x (which is a function of p and V ) ; the latter occurs only in the form sin 2 x ail( l 

 sin a 2x. Hence to integrate with respect to p we need only know the values of tin- 

 two integrals* 



4V V sin 3 \.pdp, irV sin a 2 X . p dp, 



Jo Jo 



which are functions of V only. It is sufficient, at present, to denote them by the 

 symbols Q'(V ) and Q"(V ), leaving their further consideration till later. As the 

 functional relation between x> V , an d p, and consequently also the values of 

 the above integrals, depends on the law of molecular interaction, which will difl'er 

 according as the collision in question takes place between two molecules m, two 

 molecules m', or one of each kind, we distinguish between the three cases by adding 

 the suffixes 11, 22, 12 respectively to Q'(V ) and Q"(V ). 

 Thus 



(20) A,X = 



+ V ( Y 2 + Z 2 - 2X 2 ) Q" 12 ( V )] f(u, v, w) f (u f , v', w') du dv dw du' dv' dit/. 



(21) A 12M ( a + V 2 +^) = - 



In the case of a Maxwellian gast the expressions Q ( V ) are independent of V , so 

 that the integrals just written can be evaluated in terms of mean values of functions 

 of U, V, W without any knowledge of the functions f(u, v, w), f (u r , v', w'). In 

 general, however, this is not possible ; we require some knowledge of these functions, 

 which express the law of distribution of velocity, in order to make further progress. 



4. The Law of Distribution of Velocity. 



It is well known that in a gas which has had time to attain a uniform state the 

 functions/,/' have the respective forms 



w 



k The factors 4n-V and irV are added merely for convenience. 



t Also in the case of a gas whose molecules are point centres of force which attract one another 

 according to the fifth-power law. This case, among others, is considered in Part II., and probably is 

 nearer to actual fact than MAXWELL'S case. Note added October, 1911. 



