424 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



If (j) is the angle from ki to k/ or from k4 to k/, we have from (7) 



1 —V 



in = 711 cosh — m sinh </> = m 



Vl _«2 



(20) 



where v = tanh <^ is the relative velocity of the two sets of axes. 

 But this is in fact the very relation between the energy of a given 

 particle of light as measured by two different observers whose relative 

 velocity is x. It is therefore, as far as the energy relations are con- 

 cerned, proper to consider a as a vector of extended momentum. 



The final proof of the desirability of considering the vector a as 

 extended momentum comes when we consider the interaction of a 

 light-particle with a particle of the ordinary sort. We shall see that 



the law of the constancy of extended momen- 

 tum is true, and is only true, when we include 

 the momentum of radiant energy as well as 

 that of so-called material particles. 



Let the vector a (Figure 18) be the vector 

 due to a light-particle, and b that due to a 

 material particle which has the power of absorb- 

 ing light. Then if our law of extended mo- 

 mentum applies to a and b, there will be a 

 single vector after impact equal to a -f b which 

 will represent the extended momentum of the 

 material particle after it has absorbed the light. 

 Let us choose anv set of axes. Then 



Figure 18. 



a = ai ki + 04 k4, b = hi k] + hi k4, 



where 04 = 04 is the mass of the light-particle, and 64 is the mass of the 

 material particle before impact, while (h and 61 = hi v are the respec- 

 tive momenta. The momentum after impact is 



Oi -{- hi = at + hi V. 



Hence the change in momentum of the material particle is equal in 

 our units to the energy of the light absorbed, which gives at once the 

 well known formula of Maxwell and Boltzmann for the pressure of 

 light. 



While it is evident, therefore, that such a vector a satisfies fully all 

 the conditions of an extended momentum, it must as a singular vector 

 have properties quite distinct from those of a momentum vector 

 which can be written in the form of MqW. Since a singular vector 



