the Molecular Velocities in Gases. 55 



found, if the figure be rotated one and the same angle about 

 one and the same axis carried through G r , the orientations of 

 S will again be of equal frequence. Therefore %?i"F(W / , S) 

 is a function of the same nature as /(IP), depending only on 

 the position of M" upon S', and not on the orientation of the 

 latter. We shall therefore hare 



?=V£'<KM"), 



the function <£(M") being again of the same nature. 



The sum will be formed by giving to S' the positions J)\, 

 D' 2 , . . . and taking for M", each time, the point at which the 

 normal is parallel to M'O. If we rotate all the positions thus 

 formed one and the same angle about one and the same axis 

 passing through G f , the normals to the points M" will again 

 hare a common direction, and ^(M") will be the same for each 

 figure; consequently 2/<J!>(a[") will not have changed. The 

 new positions of S' will again be of equal frequence, and will 

 then all be found to be those which it would be necessary to 

 employ on taking for M' another point of the sphere, or (which 

 comes to the same) on supposing co at another place. More- 

 over, as this may be any place whatever, the value of H is 

 independent of the position of co ; and consequently it is the 

 same with the corresponding number of collisions — which w r as 

 to be demonstrated. 



Case IV. This will be the general case, in which the mole- 

 cules will have any unequal forms whatever. 



We may evidently assimilate them to a numerous series of 

 forms F 1; F 2 , F 3 , &c, serving as types, and each of which may 

 be oriented in all possible ways. We shall divide the collisions 

 into groups by putting in the same those in which the mole- 

 cules of the first kind have one and the same figure F, aDd the 

 others one and the same, F'. From what we have just seen, 

 all directions of the new velocities will be equally probable for 

 each group, and therefore also for their union or for all the 

 collisions ; and the same property once demonstrated, equa- 

 tion (3), which is the immediate consequence of it, may also 

 be said to be so, jj, being the total number of the collisions. 

 As to this number, whatever may be the shapes of the mole- 

 cules, it is proportional to each of the quantities re, re', V, t 

 separately when it varies alone, the others remaining constant; 

 thus its ratio to nn'Yt is an absolute constant depending on 

 the nature of the molecules, and which may be designated by 

 a. Equation (1) is thus verified. 



