Complexions

From the theory of permutations we get for the number of all possible complexions:

N(N+l).(N+2) … (N+P-)_(N + P-1)
1 . 2 . 3 …P (/V-1)LP!

or to a sufficient approximation,34

__(N+P)N+P NNPP '

We perform the same calculation for the resonators of the other groups, by determining for each group of resonators the number of possible complexions for the energy given to the group.

The multiplication of all numbers obtained in this way gives us then the total number R of all possible complexions for the arbitrarily assigned energy distribution over all resonators.

In the same way, any other arbitrarily chosen energy distribution35 E, E’, E", … will correspond to the number R of all possible complexions which must be evaluated in the above manner.

Among all energy distributions which are possible for a constant E0=E+E’ + E"+… there is one well-defined one for which the number of possible complexions R0 is larger than for any other distribution.

We then look for this energy distribution, if necessary by trial, since this will just be the distribution taken up by the resonators in the stationary radiation field, if they together possess the energy E0.

The quantities E, E’, E", … can then be expressed in terms of one single quantity E0. Dividing E by N, E’ by N’, … we obtain the stationary value of the energy U„, Uv’, [/,/’, … of a single resonator36 of each group, and thus also the spatial density of the corresponding radiation energy in a diathermic medium in the spectral range

so that the energy of the medium is also determined.

Of all quantities which occur only E0 seems now still to be arbitrary. One sees easily, however, how one can finally evaluate E0 from the given total energy E„ since if the chosen value of E0 leads, for instance, to too large a value of E„ we must decrease it appropriately, and the other way round.38

After the stationary energy distribution is thus determined using a constant h, we can find the corresponding temperature i? in degrees absolute* using a second constant of nature k= 1-346 x 10"1 6 erg degree- 1 through the equation

The product k In R0 is the entropy39 of the system of resona- tors; it is the sum of the entropy of all separate resonators.

A more general calculation which is performed very simply, using exactly the above prescriptions shows much more directly40 that the normal energy distribution determined in this way for a medium containing radiation is given by the expression

which corresponds exactly to the spectral formula which I gave earlier

The formal differences are due to the differences in the definitions of uv and Ex..

The first formula is somewhat more general inasfar as it is valid for an entirely arbitrary diathermic medium with light velocity c. I calculated the numerical values of h and k which I mentioned from that formula using the measurements by F. Kurlbaum and by O. Lummer and E. Pringsheim.t