( 529 ) 
of the system of maximum frequency in an ensemble (whose modulus 
that with a fair approximation y (%,d@V») can be represented by 
Ny, 
(w, dV,) . ° ° . ; 5 . (e) 
where @, is a function of nz. I have calculated for w the approximate value. 
; Me en fay 
= — n{ — — — n’/[— 76 
og @ 5% ren 
(Cf my dissertation and also these Proceedings 1908 p. 116). 
The extension of the 37 dimensional space covered by the systems containing 
Ny... M.. Nk definite molecules in the elements dV,..dVx..dVk can now be 
represented by 
k 
IT (i dV,). 
1 
The extension covered by all possible systems of this kind amounts to 
“1 (My dV,) 
ni eae nn 
In the potential energy we may neglect the repulsive forces, these forces having 
been already taken into account by the exclusions (c). Supposing that the energy 
is the same for all the molecules of an element dV; we can represent the total 
potential energy by the formula 
k 
1 
For the frequency we find 
| 3 al NE, 
er eel k Beene 
2 7) x 4 V,, @ 
$= N(22Om) en! je 
1 Ny! 
or, introducing the function w by means of (e) 
3 = NyE, 
—n — k Ny Ee ee 
2 @ nV) g 
niee IN (pawerande 2 Ar i 
1 Ny! 
The formula (ll) is a direct consequence of the last equation. 
As we are treating a case in which there are differences in density in the 
system of maximum frequency, the question arises as to whether these differences 
have any influence on the value of the function w. If it were so, this function 
would depend not only on n, but also on the derivatives of this quantity with respect to z. 
The difference in question really has an influence on the energy, but in conse- 
quence of the hypothesis of p.p. 526 and 527 the density changes so little along 
the length dz, and the value of the exclusions at the limits of dz, is so small in com- 
parison with the value of those originating from the molecules of dz, itself, that we may 
consider q, as depending only on ny. This, however, will be true no longer if the 
sphere of action of the attractive forces is not large in comparison with g; for 
this case the following theory would have to be modified considerably. 
36 
Proceedings Royal Acad, Amsterdam; Vol. XI. 
