The Quantum of Action. 157 



(3) The discontinuous energy exchanges always occur in 

 such a way that the steady motions satisfy the equations : — 



§p a dq 2 —ah 



§p 3 dq 3 = Th 



where p, a, t, . . . . are positive integers (including zero) and 

 the integrations are extended over the values of q { , pi corre- 

 sponding to the period - . The factor A is a universal 

 constant. v% 



Here also we may represent each pair q^ pi by rectan- 

 gular coordinates in a plane, and it is a consequence of the 

 third hypothesis that the state of any system, at any given 

 time, will be represented by a point on a certain locus in 

 each q, p plane. These loci divide all the planes into equal 

 areas, 



ft dpdq = h. 



This third hypothesis is clearly very different from Ishiwara's, 

 though it is obvious that it will lead to similar and, in some 

 cases, identical consequences. The essential differences 

 between the two assumptions are : — 



(a) When a large number of systems are in statistical 

 equilibrium their representative points are, in Ishiwara's 

 theory, uniformly distributed through any region in a q, p 

 plane. These regions have not necessarily equal areas, but 

 the mean area of the regions in which the representative 

 points of any one system are situated is equal to the 

 universal constant h. In my theory each q y p plane is 

 divided into regions of equal area, and the representative 

 points of a system are always on the boundaries between 

 these regions. 



(6) Another distinction between the two theories is con- 

 nected with the limits of the integral 



$pdg. 



Ishiwara extends the integration over the whole period of 

 motion of the system, whereas in my theory it is extended 

 over the period corresponding to the coordinate concerned. 



A short critical discussion of the various forms of 

 Quantum Theory, especially those of Planck, Ishiwara, 

 Bohr, and that recently put forward by me, will not be 



