82 - MOLECULAR MOTION AND ITS ENERGY 37 



We may look on this expression as being the value of n 

 when we take the angle GAG of the same value s as in the 

 former figure. For the number which we have found for the 

 particles which fly out from A in a given direction is just 

 as great as that which we are seeking, viz. the number which 

 reach A in the opposite direction. In this respect only do 

 we alter and extend the meaning of n, that we count not 

 only the particles which move in a given direction in space, 

 but include also all those whose directions form the same 

 angle with the normal AG. 

 If we put this value 



n = 



into the above formula we obtain the number of all the 

 particles which in unit time meet the area F at an inclina- 

 tion 5, which is therefore 



cos s. 



This expression may be simplified by our replacing A cos s, 

 which represents the projection of the breadth CD = A of 

 the zone on the plane EAF, by the line HJ = H'J' = S ; 

 this gives 



for the number of particles which in unit time meet the 

 area F from the zone CDD'C'. 



From this we obtain the whole number of the particles 

 which arrive at the area F in the unit time by summing the 

 expression for all the zones, i.e. by evaluating the magni- 

 tude 



S.rS. 



To calculate the sum S.rS, let us take the zones so as 

 all to have the same value S ; we therefore divide the line 

 AF, whose length is 1, into q equal parts (where q is a very 

 large number), and put 



HJ = S = -. 



q 



Then, taking the radius r as the arithmetic mean of the 

 bounding radii AH and AJ of the zone, or 



r = (AH + AJ), 



