for the Theoretical Proof of A vogadro's Law. 309 



molecule of the first kind lies within this prism, it impinges in 

 the manner described upon a molecule of the second kind. 



Of all molecules of the first kind lying within this prism, 

 we have to consider only those whose velocities lie between 

 the limits v and v + dv, and which are moving in directions 

 making angles with the direction of V which lie between 

 the limits T and T + dT. If there are in the unit volume 

 A.irv 2 f{v)dv molecules of the first kind which fulfil the first 

 condition, then, in the above-described prism, there are 



2m?f(v)rh*T8<r . dv d& dT dO 



molecules of the first kind which also satisfy the second con- 

 dition. 



Let us further suppose that in the unit volume there are 

 47rV 2 F(V)<iV molecules of the second kind whose velocities 

 lie between the limits V and V + dV. Then, by multiplying 

 the above expression by this factor, the following expression 

 for the total number of impacts which occur in unit time in 

 unit volume between a molecule of the first kind and a 

 molecule of the second kind, so that the conditions (2) are 

 fulfilled, 



dZ = 87r 2 v 2 Y 2 f(v)F(V)r&TsadvdY dT dSdO, . . (3) 



is obtained. 



In order to obtain from this the number Z of all the 

 impacts which may occur altogether in the unit volume in 

 unit time between a molecule of the first kind and one of the 

 second kind, we have to integrate with reference to from zero 

 to 27r, with reference to T from zero to 7r, with reference to S 



7T 



from zero to », and with reference to v and V from zero 



to co ; which we may express by Z=JcZZ. 



J=j*D<iZ is the energy which is brought by all these 

 impacts to molecules of the first kind. If we integrate the 

 expression D dZ first with reference to and S, we obtain 



8^f {K ).Y^(Y)rB^dvdYdT^ R (vr 9 + ^-^ . (4) 



whilst dJL integrated with reference to the same variables 

 gives 



$7r*v 2 f(v)Y 2 F (V) r8 2 r dv dV dT. 



The quotient of the two expressions may be considered to be 

 the energy which, on the average, is transferred from mole- 

 cules of the first kind to molecules of the second kind, when 



