44(5 MR. S. CHAPMAN ON TIIH KINETIC THEORY OF A GAS 



Since, as has already been mentioned, p 2 is a small quantity of the second order 

 the term e 1 " +m ' may be neglected, since it is equal to unity to our order of approxi- 



mation. Transforming the variables x, y, z in the last integral to polar co-ordinates. 

 we have 



, \ hmm' ITffvw /v \ 



A ia w = -'- i *\ V 3 Q' 12 ( V ) e 



\tr(m + m')} JJJ 



, m' \ hmm' 



-'- i 012 



')} JJJ 



y cog x rfy g . n cQg ^ ^ ^ 



o n\ \ 



by putting x = V cos 6, &c. It must be remembered that X is a function of 9 and <j>, 

 since 



cos X = ^CJ C os 0+ ^^- sin cos + Wfl ~ W/0 sin sin . 



All the terms of the exponential series occurring in the integrand of the last 

 integral are negligible (on account of the factors /o 2 , /> 3 , and so on) except the unit 

 term and the term of the first degree. The unit term leads to a null result on 

 integration with respect to 9 and tf>. The first degree term alone contributes to the 

 final result, which (in our previous notation) is easily seen to be 



/o\ -1/2 / tn mm / , \- nl 



(32) ; ^.f^^- ( W ' - )F,, 



If we put m = m', u = u' , v = v we find that A n w = 0, as is otherwise evident, 

 since the momentum of a system of molecules is unaltered by their mutual encounters. 



Having now calculated all the values of AQ which we require, we proceed to 

 substitute them in the various special forms of the equation of transfer, in order to 

 obtain expressions for the coefficients of viscosity, diffusion, and conduction in simple 

 and mixed gases. 



7. The Coefficient of Viscosity of a, Simple Gas. 



First considering a gas composed of molecules of one kind only, we substitute for 

 A?< 2 (which in this case equals A n 2 ) and A?/v from equations (27) and (30) in the 

 special equations of transfer (ll) and (12) respectively. We get 



R// = . + _ a + _ 2 ., 



20 \Sx tiy dz/ 80: 



r/ 20 

 Remembering that = --(p zz -p), J2 = p^ (see p. 442), and comparing these 



