MATHEMATICAL INTRODUCTION 
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'■ Chapter X, p,: 
general, and tie® 
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not explained. For this reason Planck modified the exposition 
of his theory, and introduced the conception of a quantum of 
action. His statement of this new point of view, derived from 
a study of statistical mechanics and probability may be 
paraphrased as follows : 
“ The probability of a continuous variable may be obtained 
by considering independent elementary regions, of equal 
probability. ... In finding these elementary regions in 
classical dynamics, use is made of the theorem that two physical 
states, of which one is the necessary effect of the other, are 
equally probable. In a physical system, where q represents one 
of the generalized co-ordinates, p the corresponding ‘ moment ’ 
or ‘ impulse/ according to the theorem of Liouville the region 
dq considered at any instant, is an invariant with respect 
to the time, if q and p vary according to Hamilton’s equations. 
On the other hand, p and q can, at a given instant, take all 
possible values, independently of one another. Hence it follows 
that the elementary region of probability is the infinitely small 
element of the phase space dp dq. The new hypothesis has the 
effect of limiting the variability of p and q in such a way that 
these variables change only by jumps. ... In this way the 
number of elementary regions of probability is reduced, whilst 
the extent of each is increased. The hypothesis of quanta 
consists of supposing that these regions, which are all equal, 
are no longer infinitely small but finite, and that for each 
where h is constant.” 
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sed later. 
4. The Generalized Form of the Quantum Theory 
Recent progress in the development of the quantum theory 
has been based on the generalized form of the theory put forward 
independently in 1915 by W. Wilson,* and by Sommerfeld.f A 
generalization of a somewhat similar character was proposed 
about the same time by Ishiwara,t but this contained certain 
features which are not altogether acceptable. The aim in all these 
methods is to furnish a common basis on which the theories 
of full (black body) radiation, spectral series and other pheno- 
* W. Wilson, Phil. Mag., vol. 29, p. 795, June, 1915. 
f Sommerfeld, Sitz. Ber. der Münchener Akad., Dec, 1915; Ann. d. 
Physik, vol. 51, p. i, 1916. 
J Ishiwara, Tokyo Sugaki-Buturigakkwai Kizi, 2nd ser., vol. 8, No. 4, 
p. 106.