262 Free Path Phenomena [CH. xin 



If the path is small compared with the scale of variation of structure in 

 the gas, we may to a sufficient approximation write the value of B c at a 

 distance I along the path in the form 



(B c ) l = (B c ) +l( d ^} ........................ (597), 



\ 01 /o 



so that 



I 1 

 Jo 



Hence the fraction of the whole number of molecules which describe 

 the distance r from dv to dxdy without collision is 



e-r*'* .................................... (598), 



where B c is evaluated at the middle point of the path. 



316. By multiplication of expressions (594) and (598) we find, as the 

 number of particles which cross the plane 2 = z per unit area per unit time 

 with velocities between c and c + dc, having started from the element dv, 



/hs m s 



dn = dr sin cos QdOd+viJ T e- hmc * @e~ rSe c 2 dc ...... (599). 



In this expression, , v and h are evaluated at the point (r, 6, <) while B c 



(T \ 



=,6,+}. Since, however, v, h and B c are functions 

 ^ / 



of z only and not of x and y, we can write the values of v, h and B c required 

 for expression (599) in the forms 



86 



= r cos 6 -=- , 



a dv 



v = v r cos 6 , 

 oz 



77 a dh 



fi = h r cos 9 ;r- , 

 oz 



in which all the quantities on the right-hand side are evaluated in the plane 



Z = Z . 



Substituting these values, we obtain as the value of dn 



/h?m s 

 dn = dr sin cos 0d0d$v A / - r- eH" 7 * 2 e- rBc c 2 c?c 



V "n 



1 r cos ^ -fr -= r cos ^ - -r- 



<s) a^ v 



a a a ] 



%r cos T -=- + r cos me 2 -y- + *r 2 cos - T - , 



\ 



dh 



- -y- - T 



h dz dz dz 



in which all the quantities are evaluated in the plane Z = Z Q . 



