CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 445 



6. Calculation of A w . 



Finally we turn to consider A,.,, the calculation of which is rather different from 

 that of the foregoing quantities because ?/ and ?/, ?' and t'' 0) .' and / are now 

 different. We require A ia tt in order to determine the coefficient of diffusion. 



It is evident from equation (2) that 



rS 13 ii de = 





-, ('*')> 



^o IHTH 



so that 



/ fft 



m + m' 



A u w = W/-J2 , fff fff Q' ta (V u )(n'-u)S(u,v,w)j'(u',tf,u/)dudvdwdu'dv'fl'. 

 m-\-m JJJJJJ 



In the present case we may neglect the small deviation from MAXWELL'S law of 

 distribution of velocities, and thus avoid making any assumption as to the form of F. 

 This is so because in this case (unlike all the preceding cases considered) the unit 

 term of (l + F+F') in the expression for ff leads to a term in the final result, so that 

 the terms which would result from the inclusion of F and F', being multiplied by the 

 small coefficients a and ', are negligible to our order of accuracy. The term arising 

 from the unit in (l + F + F') is itself a multiple of ' , which we assume to be so 

 small that its squares and higher powers are negligible ; u' u a being thus a small 

 quantity of the first order, the terms arising from F and F' are of the second and 

 higher orders of small quantities. 



Thus we now have 



to / 



h. mm 



's <->) 



i2v>y/\ :t/ -' 

 e" 



where we have written (x, y, z) for (u u', vv', wtt/), so that V a = 

 Changing the variables in the above integral to (x, y, 2) in place of n', t/, /, and 

 recalling that 



f< 



J -go 



\i/a 



-nP+M^ = (- I " K>l4a - 



w 



on integration with respect to , i', ?// we obtain 



/ m' f hmm' 



A u n = - 



i hmm' l ?/a fflV^ nr \ -*^,<v,,'-a>Voeo. *+,.) , , , 

 { x\ i/ ia ( V ) xe *-' dx dy d 



[ir(m+m)i JJJ 



m + j/i 

 where we have written (V,, 2 2pV cos \+p 3 ) for 2(x+tt' ) 2 ; thus 



-tt' ) a , pV l) coB\ = x(u' -u l) )+y(v' v )+z('u/ () -iv l) ). 



