Richard Chace Tolman

“ The special purpose of this note is to make clear that the fifth fundamental equation of electromagnetic theory may be derived from the four field equations and the principle of relativity without making any arbitrary convention as to the mass of a moving body. It is quite unnecessary to place the transverse mass of a moving body equal to and the longitudinal mass equal . ”

“ If E is the force acting on a small stationary test charge of magnitude , then E' will be the force acting on the same test charge or electron when it is moving through the point in question with the velocity , the force E' being measured in units of the moving system [5] .
We are more particularly interested, however, in the vector F which determines the force F that nets on the moving charge but which is measured in "stationary units," thus determining the equations of motion of the test charge with respect to stationary coordinates. ”

“ Returning now to the consideration of an electron which is moving with the same velocity as the system S', we see that the transformation equations (13-14-15) together with the above equations lead to the relation
which is the desired equation:
This result agrees with that obtained by Einstein in his second treatment (Jahrbuch der Radioaktivität, iv. p. 411, 1907) , where instead of defining force as equal to mass times acceleration, he defined it by the equations
which agree with our definition of force as equal to the rate of increase of momentum. ”