( 319 ) 
where 
y= dan fp. 
Now we are going to seek the probability that the total moment 
M of the volume element i.e., the sum of the amounts contributed 
by the separately considered groups of molecules, has amplitudes 
which lie between: 
[Mo] and [Ma 4+- dM oa]. 
According to the calculus of probabilities the probability for such 
a sum is again represented by a function of the same form as the 
separate terms, while the modulus is the root of the sum of the 
squares of the moduli of the separate terms. So: 
C= vy (LF a®, Fdw dr, 
If we take the integrals between the limits — oo and +, the 
factor 4 must not be omitted, because we have to take ie half 
of all the groups: for if we ake a group with definite amplitudes, 
that one with equal amplitudes, but of opposite sign has been taken 
into account at the same time. 
For the other quantities M,o, My, My2, Ma and Mo of course the 
same formula holds good. Now we have still to prove that the 
chances for these quantities are independent of one another. To 
this purpose we draw vectors from point O, which have the 
quantities Mar, My, and Mz: as components. Along the axes the 
density of the final points of these vectors is the same as in the 
distribution of velocities of MAxwrELL. If a large M‚j was probably 
accompanied by a large M,, the distribution in space would not be 
that of MaxweLL. The choice of the axis is however perfectly 
arbitrary, and the distribution along every line passing through 0, 
must be the same as along the z-axis. From this follows that the 
distribution is really the same, as that which MAXWELL found for 
the velocities, i.e. that the chances of the quantities Mei, My and 
M., are independent of each other. In a corresponding way we may 
prove this for Mm en Mo, My and Ms, Mi and Mas. 
If we represent the mean of the squares of all quantities ax 
by «2, we get: 
d= V dn. Wan 
and 
ne 
Go 
Proceedings Royal Acad. Amsterdam. Vol. LI. 
