180 
The g-point of the system in the six-dimensional u-space (q,, . . . p‚) 
is then limited by the quantum-hypothesis in the following manner: 
its projection on the plane g,, p, must fall on one of the Pranck- 
ellipses; similarly the projections on the planes q,, p, and q,, p,, 
If the total energy EH is contained between 0 and a moderate value, 
we see by (65) that g,,p, may still fall on a large number of diffe- 
rent ellipses; (since for this degree of freedom the energy-stages 
&, = 0, hv,,2hv,,... follow each other closely), g,,p, on the other 
hand only on a few ellipses, whereas q;,p, is possibly completely 
confined to the position g, = p, = 9. 
If the limitation which is due to the quantum-hypothesis did not 
exist, the “weight” to be given to a given region in the u-space 
would according to BorrzMANN simply be its volume 
eae dp. TLE GN 
To each region, whose three projections are three PLanck-ellipses, 
we assign the weight 
RE DN 
The joint weight of all phases which the u-point can assume, 
when the energy is subjected to an upper limit, will then be 
fyizc SSS... ee 
nn 7% 
where the summations are to be extended over all the quantum- 
numbers which the first, second and third degree of freedom can 
assume. With a moderate upper limit for the energy t, as we saw 
would be able to rise to high values, and the corresponding sum 
may accordingly be replaced by {fea dp,; t, on the other hand 
would be confined to zero and the corresponding sum reduce to the 
first member /. Hence 
eee [ta dey (EIA -. 
According as the upper limit for the energy is made to rise or 
fall (i.e. the second degree of freedom is made to pass from the 
state of being half-excited to that of full excitation or non-excitation) 
(69) changes into 
gen ld dp, da dp zt … far, dp, . (70) 
Lr fs op, neren far dp, ee 
or 
ate t—“‘“SCS:;~‘<‘;..”.”;”tt 
