for the Theoretical Proof of Avogadro's Law. 329 



polar coordinates of the point H of the sphere, we have 



The left side of equation (33) is next transformed into 



d£d v dZdZ 1 dr )l d£ l vdNd'E (34) 



If, now, we denote the projections of the relative velocity 

 XlRi before impact on the axes of coordinates by x, y, z, and 

 also the projections of the relative velocity HR/ after impact 

 on the axes of coordinates by x', y', z r , and with constant f , ??, f 

 introduce the variables 



*=&— fc y-vi—n, 2=£i-fc 



expression (34) becomes 



d^drjd^dxdydzvd^dE. .... (35) 



Then we leave x, y, z, N, E constant, and instead of f, r), J 

 introduce the variables f ', 77', f. If x, y, z be the projections 

 of the line E x R/ of the relative velocities drawn from fl on 

 the axes of coordinates, we have 



J^_ • M yi _ Mgj 



?_? ro + M' ^ ^ m + M 5 ^ ~^ m + M 



Since, now, all the lines drawn in fig. 3 remain altogether 

 unaltered in magnitude and position, x 1} y 1} and z 1 are also 

 constant, and we have 



dgd v 'dg=d%d7idZ. 



Hence expression (35) becomes 



df-'drj'dgdadydzvdNdEi (36) 



The next step consists in introducing for constant {', 77', £', 

 N, E the variables x', y', z' instead of x, y, z ; that is, the 

 coordinates of the point R/ instead of the coordinates of the 

 point R r It is at once seen from tig. 3 that the element of 

 volume described by the point Rj on change of its coordinates 

 is exactly equal to that which the point R/ describes for the 

 position of the point H remains unchanged. It follows, 

 therefore, that 



dxdydz=dx dy' dz' ; (37) 



and expression (36) becomes 



dgdv'dgdx'dy'dz'.v.dN.dE. . . . (38) 



Now, again, inversely 



?/=?'+*', ni'=v'+y', &+■?+* 



