324 I)R. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
In the case of molecules which obey the fifth-power law, therefore, the velocity 
distribution function has the simple finite form (cf. (142)) 
(198) /(U,V,W) = 
7T 
e 
-hm( U 2 + V 2 +W 2 ) 
1 “(nA ,)- 1 {V f ( U |f + V |f + ) U 1 +l2hmV) 
+f (2 Jim) (c n U 2 + c 22 V 2 + c 33 W 2 + 2c 23 VW + 2c 31 WU + 2 c 12 UV)j 
(Maxwellian molecules), 
where C 2 = U 2 + V 2 + W 2 , c u , c 12 , &c., are given by (72), and (cf (183)) 
(199) 5 Aj = 10 (|-Km) 1/ ' 2 [ {l—P 2 (cosx)} a da — ^(Km) 1/8 [ siifiy • a da. 
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poo 
M axwell # calculated the value of the integral sin 2 y . «■ da, the forces being 
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repulsive, by numerical quadrature, and found that 
poo 
7T sill 2 x . ada = 1'3682, 
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so that, for repulsive forces proportional to the inverse fifth power of the distance 
( 200 ) 
5 Aj = — U3682 (Km) 1/2 , 
2 
where K m 2 is the force between two molecules at unit distance. 
Molecules which are Rigid Elastic Spheres. 
§ 9 (D) We next consider molecules which behave at encounter like rigid elastic 
spheres of radius a-. This particular molecular model has been more used than any 
other, in researches on the kinetic theory, on account of its simplicity and concreteness, 
which aid the imagination in following or constructing “ descriptive ” theories of 
gaseous phenomena. As regards the analytical development of the theory, also, it is 
probably the simplest case after that of Maxwellian molecules. The difference 
between the two models in this respect is, however, enormous, the rigid elastic 
spherical molecule requiring the infinity of terms /3 r , y r in the velocity-distribution 
function, just as in the case of the most general molecular model. The comparative 
simplicity of the present model lies in the moderately tractable expressions for h rs , c rs 
to which it leads. Apart from the methods of the present and my former paper, 
* Maxwell, ‘Collected Papers,’ ii, p. 42. His constant A 2 equals it sin 2 x • ada. in our notation. 
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