2 S. Chapman — Kinetic Theory of Gases 



In this expression the upper Hmit y)o is the (unique) positive 

 real root of the equation 



(2.4) 1-7)2- ~-(^y = o 



n — I \ a / 



When n = 5 the expression (2.3) for 6 is an elliptic function, 

 and its numerical value as a function of a can be obtained 

 from tables of elliptic functions, lysine: these values for sub- 

 stitution in (2.1) and (2.3), Maxwell numerically integrated the 

 latter, and obtained the results 



(2.5) 1,(5)= 2-6595, I,(5)= 1-3682. 



Another value of n which gives easily calculable results is 

 11 = 2; in this case it is readily found that 



cos-f)= 5- 



I +a^ 



where a^ is the upper limit in the integrals Ii and I2 ; 

 physically regarded this limit cannot now be taken infinite, 

 since then Ij and L would also become infinite. The value 

 of ao to be adopted depends on the maximum distance apart 

 of the lines of relative motion of. two molecules during a 

 binary encounter. The case n = 2 is of physical interest only 

 in relation to the extreme conditions found in the interior of 

 stars, when the gas-particles are highly ionised. 



Another partly integrable case is that of n = 3, for which 

 it is easilv seen from (2.3) that 



It is convenient to change the variable in (2.1) and (2.2) to Q, 

 using the equations 



ada = ida^- = (ix)^ ( (iTu)^ - 6^ I -2 q d 6, 



the limits of 6 being o and ^ %. The integrals (2.1) and (2.2) 

 are readily obtainable by numerical quadrature after this 

 substitution, the integrand being finite within the range of 

 integration and at Q = o, and having a finite limit at e = -|x. 

 The values obtained, dividing the range of integration into 20 

 equal intervals, and applying Weddle's rule for 11 ordinates 

 (repeated), are as follows : 



Ii(3) = 5'099- 10(3) = 3*823. 



For ordinary gases the values of n which are of interest 

 range from 5 to about 15. 



(3) To determine I^ and L for general values of n, it is 



