315, 316] Conduction of Heat 263 



We next integrate this expression with respect to 6 and <f> so as to obtain 

 the total number of molecules which cross the plane z = z from the side z < Z Q 

 with velocities between c and c + dc, coming from a distance intermediate 

 between r and r + dr. In this integration, the limits for </> are from to 2-7T, 



*7T 



those for 6 are from to - only. If we denote the number in question by 



dn r>c , then, since 



re-v/z r<t>=%Tr 



sin 6 cos 9 d6d<f> = , 



J = J$ = o 



rd = ir/2 r$ = 2j 

 J 0=0 J <f>=0 



we have 



dn r>c = 2 2 



0=0 <= 



[^ 

 *~ 



7T 



r 



r 2 c 



(600) ' 



We can now sum this number over all possible values of r. The limits 

 are of course r = and r = oo . The number obtained, which we can denote 

 by dn c , will be the total number of molecules crossing the plane z = z per 

 unit area from the direction z < z , with velocities between c and c + dc. 



Since 



1 



L 



o 



/> 



Jo 



/: 



we obtain, on integration, 



dn c = 



j-=oo 



7T 3 



1 1 cZ@ 1 dv 



dz %hBc dz 



mc^dh J^dBc'] ...(601). 



T aflL f d* T aiV d~ 



