510 SECTIONAL COMMUNICATIONS. 
and therefore this ‘€-path’ comes arbitrarily close to all points of a 
u-dimensional region, while the phase path itself comes arbitrarily close to 
all points of the p-dimensional region Go. We should now have to consider 
more closely the correspondence between these two regions, using the 
property remarked on at the end of §5*. If we could succeed in showing 
that this correspondence is one-to-one and continuous, the ‘u=p-conjecture’ 
would be proved, as it is known that in such transformations the number 
of dimensions is preserved.’ 
(6) As it is still uncertain whether the “uw=g-conjecture’ is correct, 
the following remark on its relation to the quantum theory must suffice 
for the present. If in the sense of the above explanations we did succeed 
in designing systems which possess multiple periodic motions with u 
independent frequencies where s <u < 2s—1,the quantum rules as we know 
them could not be applied to such systems, for those rules are restricted to 
systems for which uZs.°* 
4 The following example at first sight against the ‘w—p-conjecture,’ shows 
how important it is to pay heed to this property. Let g=cos 9(t) and p= sin 
@ (t) where @ (¢) is expansible in multiple Fourier series, for example with w=3. 
The (q,p) point remains the whole time on the circle g*+-p°*=1. Thus e=1 and u=o. 
But here we also have violation of the condition that q¢ p, ¢ #, etc., always return © 
to their initial values when q and p do so, as must be the case for a system obeying 
Hamilton’s equations with a time-free H (g, p). Starting from this remark it is easy 
to see clearly how the ‘ quasi-periodicity ’ of the motion imposes very strict relations 
between the Fourier expansions of the q’s and p’s, and those of all their time derivatives 
on account of their common periodic character. 
5 L, E. Brouwmr, Math. Ann. 70, p. 161 (1911); 71, p. 305 (1912); 72, p. 55 
(1912). 
6 In an extremely simple case of one degree of freedom an ‘ excess of frequencies ’ 
of somewhat similar though really different kind has been treated with the helps of — 
the correspondence principle by P. Exrenrest and G. Breit [Proc. Amsterd. 25, p. 2, 
1922—Zsch. f. Phys. 9, p. 207, 1922]. Cf. N. Bour, Zsch. f. Ph. 18, p. 147, 1923. 
7 Bong has given his ideas about those cases, in which a multiple periodic motion — 
is not to be expected, in Zsch. f. Ph. 13 (1923) [p. 134—particularly in the comment — 
on A. SmeKat, Zsch. f. Ph. 11, p. 294, 1922], and also about those cases in which the — 
motion can be very closely represented by a w-multiple periodic motion with ws — 
[in *‘ Quantentheorie der Linienspectra (1918).’ Vieweg 1923, pp. 69, 70, 134]. But 
see particularly Bohr’s remarks on the failure of the quantum theory of multiple- 
periodic systems and how it shows itself in some problems of the complex structure of 
spectral lines, and of the anomalous Zeeman effect [Ann. d. Phys. 71, pp. 275-277, 1923]. 
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