Partition of Energy between Matter and Radiation. 17 



plausible. Although we cannot' deal with single molecules, 

 it seems impossible short of that to fix any limit for the 

 initial distribution. This can hardly be interpreted in terms 

 of dynamics, save by assuming that any values of the co- 

 ordinates may occur. But (2) is certainly not always true 

 when n is infinite. Thus th« vortex theory of matter allows 

 only such values of the variables as sati-fy the condition of 

 constant circulation ; for that is the essence of the distinction 

 between matter and what is not material. If for any reason 

 (2) be abandoned, it need not be concluded that there is no 

 state of temperature equilibrium. We infer rather that the 

 theory of heat depends upon properties of matter more special 

 than the abstractly dynamical, an inference which is in itself 

 made certain by the observed laws of radiation. These involve 

 an absolute constant, a length in dimensions, which cannot enter 

 into the purely dynamical scheme (Jeans). The conclusion 

 thus forced upon us ought to be as welcome to the physicist 

 as it is distasteful to the mathematician. A deduction of 

 Wien's law from Hamilton's equation adds nothing to our 

 knowledge of the nature of matter ; an explanation of it 

 may yet add much 



§ 3. The Partition of Energy in a EUmiltonian System. 

 The equations of motion are 



dt dp r ' dt dq r V ' ' ; v J 



The kinetic energy belonging to the rth degree of freedom 

 I define as equal -to 



iP--Jp r (2) 



This agrees with the ordinary definition when II is 

 quadratic. 



Notice also that if L is the Lagrangian function, we 

 have 



*«*4£-*i- • • - • • (3) 



l(L+H)=Zi Pr ?g (4) 



L is in the simple case equal to the difference of the 

 kinetic and potential energy ; H is equal to their sum. 

 Thus the kinetic energy again appears in (4) as a sum of 

 terms of the form (2). 



We write 



dN = dp Y dp 2 ... dp n dqi dq 2 . . . dq n . . . (5) 

 Phil Mag. S. 6. Vol. 21. No. 121, Jan. 1911. C 



