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III. Hamilton s Equations and the Partition of Energy 

 between Matter and Radiation. By S. B. McLaren, 

 Assistant Lecturer in Mathematics in the University of 

 Birmingham *. 



§ 1. Introduction. 



IN this paper two things are done. Maxwell's theory o£ 

 partition is extended to forms o£ energy not quadratic 

 in the velocities or momenta, and it is applied to the inter- 

 action o£ matter and radiation. As to the form, it is 

 enough if we are assured that for finite energy all momenta 

 are finite. That condition is satisfied by the expressions for 

 energy which arise in the electromagnetic theory of mass, 

 by the formula c(jt9 2 + ??? 2 c 2 )% for example, which gives the 

 energy of Lorentz's deformable electron. There p is the 

 momentum, m the mass for an infinitesimal value of p, and 

 c the velocity of light. 



For the rest (§ 4), I have tried to fill the gap which 

 Larmor (Bakerian lecture) has remarked in the work of 

 Jeans and Lorentz on radiation. With Jeans the radiation 

 is confined to a finite space bounded by reflecting walls. 

 Since within these there is no ordinary matter, all radiation 

 falls at once into the normal vibrations proper to the space 

 enclosed ; no redistribution of energy is possible, and the 

 amount of energy to be assigned to each normal mode is 

 fixed once for all by applying* Fourier's analysis to the 

 original field of force arbitrarily given. 



if Jeans can nevertheless draw conclusions as to the 

 partition of energy, it is because he assumes that there 

 is one dynamical system of which matter and asther are 

 parts, that Maxwell's statistical method can be applied to it, 

 and that in any complete formula for the energy his 

 expression for the radiation will form a part. It is such 

 a formula I give here ; but I have not reached it except 

 by assuming the atomic structure of matter. My electron 

 is an invariable distribution of charge free only to move as 

 a whole. It then appears that the whole electromagnetic 

 energy must be regarded as belonging to the radiation, 

 excepting only a term which depends on the position of 

 the electrons at any instant and would be the electrostatic 

 energy if they were at rest. In this division nothing is 

 left for kinetic energy of the electrons, the system of 

 equations cannot be brought to Hamilton's form, and it 

 does not seem that Maxwell's statistical method can be 

 * Communicated by the Author. 



