270 Free Path Phenomena [CH. xiv 



and as far as the first order of small quantities, 



/ r* : j 



e- hm ^*\Nd\N = e- hm ^wd\N = -^r, 

 -> Jo 2hm 



r<*> / / 



/ e~ hm ^ w d\N = w e ~ hm ^ d\N = %w \/T 



J -w Jo V Am 



Hence the value of expression (622) is 



(623). 



The second integral required for the evaluation of expression (620) is 

 ft*/*. V Tr 3 JJJ 6 cos w u v w ( ). 



Since X is a small quantity of the first order, we may put w = under 

 the sign of integration, and if we now use the transformation 



u = c sin 6 cos <f>, 

 v = c sin 6 sin </>, 



w = c cos 6, 

 the expression 



Bi*! 



= x^ 



Va*/ z . v ^ ^_ o 9 o _ ( 



/a^\ Tito? i i 



S/V ^ 3 ' 2/tW 3' 



,..(625). 



By subtraction of expressions (625) and (623), we find as the total number 

 of molecules of the first kind which cross the plane Z=Z Q per unit area per 

 unit time in the direction of z increasing (expression (620)), 



(626). 



(627). 



\lirhm 

 Similarly the number crossing in the reverse direction is 



The rate of increase of molecules of the first kind on the positive side of 

 the plane z = z per unit time per unit area is the difference of these two 

 expressions, and therefore denoting this quantity by I\, 



r/ \. *.' *1 i^ V T\ /COQ\ 



1 = w yiW- -T U- ........................ (628). 



