534 Dr. H. Stanley Allen on 



III. Generalized Statement of the Quantum Theory. 



13. The general form o£ tb.3 quantum theory put forward 

 by W. Wilson is based on hypotheses which may be sum- 

 marized as follows. During certain intervals each dynamical 

 system behaves as a conservative one, and between these are 

 relatively very short ones, during which definite amounts of 

 energy may be emitted or absorbed. The motion of a 

 system in the intervals between such discontinuous energy 

 exchanges is determined by Hamiltonian dynamics as applied 

 to conservative systems. Let q u q 2 , ..., p\, p 2 , ... be the 

 Hamiltonian positional and impulse co-ordinates of a system 

 in one of its steady states. The kinetic energy, T, expressed 

 as a function of q 1} q 2 , ... and q u q 2 , ... is homogeneous and 

 of the second degree in q x , q 2) .... If T contains products 

 q n q s (rz£s), such terms may be removed by a suitable sub- 

 stitution, so that the kinetic energy may be assumed to have 

 the form 



T = iA 1 j 1 2 + iA 2 g 2 2 +... 



Further, 



= TH-T 2 + (17) 



and consequently 



2T l = ^iPl> 2T 2 = q i p 2 , ..., 



so that 2§T 1 dt= §pidqi, and so on. 



It is assumed that the system in one of its steady states 

 has a period l/v l corresponding to q^ a period l/v 2 corres- 

 ponding to q 2 , &c. Wilson's final hypothesis is that the 

 discontinuous energy exchanges always occur in such a way 

 that the steady motions satisfy the equations 



2§T l dt=§p 1 dq 1 = n 1 h^ 



2§T 2 dt=$p 2 dq 2 = n 2 h[> ' • • ' ( 18 ) 



where w 1? n 2 , ... are positive integers (including zero) and 

 the integration is extended over values corresponding to the 

 particular period considered. 



Now T x represents the instantaneous value of the part of 

 the kinetic energy corresponding to the co-ordinate q x . The 

 time average of this part of the kinetic energy, estimated 

 over the period 1/v, is j^j T 1 ^==-Jn 1 /z»/ 1 . Let this time 



