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It is evident that these equations (1a-4a) must be true no matter what the velocity between the original system of coordinates and the new observer, that is they are true for all values of $\phi$. The velocities $u_{x},u_{y},u_{z},v_{x}$, &c, are, however, perfectly definite quantities, measured with reference to a definite set of axes and entirely independent of $\phi$. If these equations are to be true for perfectly definite values of $u_{x},u_{y},u_{z},v_{x}$, &c., and for all values of $\phi$, it is evident that the function $f()$ must be of such a form that the equations are identities in $\phi$. As a matter of fact $\phi$ can be cancelled from all the equations if we make $f()$ of the form ${\tfrac {1}{\sqrt {1-{\frac {(\ )}{c^{2}}}}}}$; and we see that the expected relation is a solution of the equations. Although this does not exclude the possibility that there may be other solutions of these functional equations, nevertheless from a consideration of the complexity of the equations it appears doubtful if any other simple function would satisfy the necessary requirements.

In conclusion it is to be noted that in these derivations no reference has been made to any electrical charge which might be carried by the body whose mass is to be determined. Hence, if these considerations are correct, we may reject the possibility of explaining the Kaufmann-Bucherer experiment by assuming that the charge of a body decreases with its velocity^[1], since the increase in mass is alone sufficient to account for the results of the measurements.

Cincinnati, Ohio.

October 31, 1911.

1. The possibility of explaining the Kaufmann-Bucherer experiment by assuming that the electrons have less charge at higher velocity was suggested by Professor More of this University: Phil. Mag. xxi. p. 196 (1911)