12 Mr. S. B. McLaren on the Emission and 



But to go beyond this point is to meet the great stumbling- 

 block in the way o£ all theories of radiation. We may 

 construct laws of force which make the emission a function 

 of \6, and so treat the problem as an exercise in the 

 dynamics of a particle. To proceed thus is to ignore the 

 fact that on all dynamical principles the absorption is a 

 function of the same type as the emission and that the final 

 conclusion is only the unwelcome paradox of equipartition. 



§ 4. The Disturbed Orbits. 



In what follows I treat the electron as a dynamical 

 particle moving in a conservative field of force. The equa- 

 tions of motion used are Hamilton's, and the momentum may 

 depend in any way upon the velocity or even upon the nature 

 of the field. The vector notation is employed throughout. 

 The problem is to find the deviation produced by the 

 radiation. 



Letr=(^, y,z) denote the vector whose components are 

 the coordinates of the electron at time t, supposing its orbit 

 undisturbed by radiation. 



dx 



u =df 



D = (f, r} } f) is the displacement due to radiation. 



p = (p t q, r) stands for the momentum of the undisturbed 

 motion. 



P = (A,, fi, v) the increase of momentum due to the radia- 

 tion. 



Thus v + D and p + P are the position and momentum of the 

 electron in the actual motion. 



The equations of motion of the electron written in Hamil- 

 ton's form are 



£i> + V,.ff-0 (7) 



at 



|"V^=0 (8) 



These are for the undisturbed orbit. Vr^ an d Va^? 

 indicate the vectors whose components are 



dH dH dH and dH dH dH 



dx* dy^ dz dp 9 dq' dr° 



The mass of the electron may depend in any manner 

 whatever consistent with (7) and (8) both upon its velocity 

 and the external field. i7is the total energy. 



Let W be the electric intensity on the moving electron 

 due to the radiation. To find the deviation due to E', write 



