ON THE PRINCIPLE OP RELATIVITY. 243 



If we choose to call the vector 



mw 

 s/(l-wyc*) 



the momentum of the particle and to call 



the jforce acting on the particle, the above equations express that the 

 force is equal to the rate of change of the momentum, but the momentum 

 is not proportional to the velocity unless we neglect quantities of relative 

 order (w 9 /c"). 



We may from these equations obtain another 



(J ( _ w ° 2 ) _ v dx ih/ dz 



dt X v/(i _ w,2/c 2 i - ** dt + K * dt + lv = dt 



where K is the vector above called the force. 

 If, again, we choose to call 



mc 1 



S{l-w*/c*) 



the energy of the particle, we have a generalised energy equation, and 

 neglecting higher powers of w/c the energy is mc 2 + \ mw' 2 which differs 

 from the energy of Newtonian theory only by a constant. No hypothesis 

 has been made here as to the nature or origin of m. 



We may consider a little more closely the question of the so-called 

 electromagnetic momentum and energy. Einstein 5 considers a system 

 subject to no external constraint such as we might imagine an electrically 

 constituted atom to be, its configuration being determined by internal 

 action alone, and, supposing it to take up energy from incident radiation, 

 shows that the electromagnetic energy taken up when the system is 

 supposed to be in motion is /2 times the energy taken up when it is 

 considered to be at rest. Supposing that part of the energy is given up 

 to moving particles after the fashion just described, we have exactly 

 a similar result for the corresponding amounts. 



Thus we have the result; — 



(m + E> 2 



and further it appears that, if a similar calculation is made for momentum 



_ (m + E )w 

 ^(1 -wye-')' 



Thus it would appear that, as far as the mechanical equations go, it is 

 only (m -J- E„) that matters, and m might even be zero. 



s Jahrb. d. Radioakt. und Elektr., i, 1907. 



M 2 



