WILSON AND LEWIS. — RELATIVITY. 421 



By comparison with equation (9), or by substituting for m from (11) 

 and differentiating, we obtain the results^ ^ 



(15) 

 (16) 



where dE/dt represents the rate at which energy is acquired by the 

 particle when acted upon by the force/. Since dE/dt and dm/dt are 

 equal, we may, except possibly for a constant of integration, write 

 E = m. This is a special statement which falls under the more 

 general law, that the mass of a body, in the units which we employ, 

 is equal to the energy of the body. We may therefore use the terms 

 mass and energy interchangeably. 



The type of motion which, from the viewpoint of the principle of 

 relativity, corresponds most closely to motion under uniform accelera- 

 tion in Newtonian mechanics, is motion under a constant force /. 

 The equation of motion may readily be integrated. 



.dmv d V „ 



V _ K , s dx _ Kt 



\t — foj, 



V(l_t,2) m " dt ylm^^-\- K^it — toY-' 



and \ ~ ^^ '^ KJ ~ ^^ ~~ *^^' ^ W' 



The representative point in the .r^-plane therefore describes a pseudo- 

 circle of which the curvature is the constant force acting on the particle 

 divided by wo. The mass of the particle at any time is 



m = 



which shows that the increase in mass is equal to the product of the 

 force by the distance traversed, as it should be from the principle of 

 energy above stated. 



23. Let us consider the problem of the impact of two particles A 

 and B of which the vectors of extended momentum (tmyr) are respec- 



21 See later discus.sion (§36) of the so-called longitudinal mass. 



