# Page:CarmichealMass.djvu/11

which we proved theorem II. As before the balls will appear to B to move on arcs of the circle

${\displaystyle x'=\cos \theta ,\ y'=\sin \theta .}$

Suppose that to A they appear to move along arcs of the ellipse[1]

${\displaystyle x=\cos \theta ,\ y=\rho \sin \theta }$

where ρ is a constant to be determined. As before, without the use of postulate R, it may be shown that to A the periods will be the same in the two cases. Then determine the periods as in the preceding discussion. The expressions for the period will contain ρ; in fact on equating them we shall find

${\displaystyle m_{2}=\rho ^{2}m_{1}.}$

But ${\displaystyle m_{1}=l\left(m_{v}\right)}$ and ${\displaystyle m_{2}=t\left(m_{v}\right)}$ whence on account of the relations between ${\displaystyle l\left(m_{v}\right)}$ and ${\displaystyle t\left(m_{v}\right)}$, we have at once

${\displaystyle \rho ={\sqrt {1-\beta ^{2}}}}$

This, in connection with postulate L, leads readily to the usual relations concerning the units of length in two systems of reference. Having these relations of length units, the dimensional equation

${\displaystyle V={\frac {L}{T}}}$

taken in connection with postulate V leads at once to the usual relation of time units, provided we take the motion along the line of relative motion of the two systems. Hence we have the following theorem:

Theorem VI. If ${\displaystyle l\left(m_{v}\right)}$ and ${\displaystyle t\left(m_{v}\right)}$ have the same meaning as in theorems L and II. and if for any particular kind of matter whatever we have the relation

${\displaystyle t\left(m_{v}\right)=\left(1-\beta ^{2}\right)\cdot l\left(m_{v}\right)}$

then this fact and postulates ${\displaystyle \scriptstyle \left(MVLC_{1}C_{2}\right)}$ are essential equivalents of postulates ${\displaystyle \scriptstyle \left(MVLRC_{1}C_{2}\right)}$.

## § 5. The Bucherer Experiment.

[2] Our postulates V, L, ${\displaystyle C_{1}}$ have been universally accepted as part of the basis of the classical mechanics. Many persons have found no difficulty in accepting postulate M; certain it is at least that we have absolutely no evidence to contradict it. We have seen in theorem V, that these four postulates, taken in connection with the formula for transverse mass, form

1. Since we are assuming postulate L it is clear that the path must be of this form.
2. Compare Tolman, Physical Review, 31 (191 o), p. 36.