# Page:CarmichealMass.djvu/14

quantities ${\displaystyle e/t\left(m_{v}\right)}$ and ${\displaystyle e/l\left(m_{v}\right)}$, where ${\displaystyle t\left(m_{v}\right)}$ and ${\displaystyle l\left(m_{v}\right)}$ denote as usual the transverse mass and the longitudinal mass respectively of the moving electron, whose velocity is v. Bucherer's methods furnish the means of measuring the first of these ratios; it will be necessary to devise a way to determine the value of the second ratio.

Or, instead of finding a means of measuring the two quantities ${\displaystyle e/t\left(m_{v}\right)}$ and ${\displaystyle e/l\left(m_{v}\right)}$ it will be sufficient if one determines only their ratio, as will be obvious from the discussion following.

Suppose now that we find the relation predicted by the theory of relativity:

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

This equation leads to the relation ${\displaystyle t\left(m_{v}\right)=\left(1-\beta ^{2}\right)\cdot l\left(m_{v}\right)}$. According to theorem VI. this would give a new experimental confirmation of the theory of relativity. The importance of such a result is apparent.

But we should also have more than this. Having now concluded that the theory of relativity is confirmed and this result having been reached without the use of a relation between ${\displaystyle e_{0}}$ and ${\displaystyle t\left(e_{v}\right)}$ we may now use the experiment of Bucherer to draw further conclusions concerning electric charges in motion. In particular, it is obvious that we should have a proof of the fundamental relation

${\displaystyle e_{0}=t\left(e_{v}\right)}$

That is to say, having assumed that ${\displaystyle t\left(e_{v}\right)}$ and ${\displaystyle l\left(e_{v}\right)}$ are equal we conclude further on experimental evidence that each of these is equal to ${\displaystyle e_{0}}$. Now it is difficult to conceive how ${\displaystyle t\left(e_{v}\right)}$ and ${\displaystyle l\left(e_{v}\right)}$ could be different, for this would imply that the notion of electric charge is in need of essential modification. In fact, if the charged body is moving, the notion of charge would be indefinite in meaning until we had assigned the direction along which such charge is to be measured. Thus, if the experiment should turn out as surmised above, we should not only have the strongest sort of experimental confirmation of the theory of relativity but we should also have a valuable verification of the fact that an electric charge does not vary in amount with the velocity of the body which carries it.

Suppose, on the other hand, that we make no assumption concerning the relation of ${\displaystyle t\left(e_{v}\right)}$ and ${\displaystyle l\left(e_{v}\right)}$ or of ${\displaystyle t\left(m_{v}\right)}$ and ${\displaystyle l\left(m_{v}\right)}$. On carrying out the experiments a relation of the form

${\displaystyle {\frac {t\left(e_{v}\right)}{t\left(m_{v}\right)}}=k{\frac {l\left(e_{v}\right)}{l\left(m_{v}\right)}}}$

will be obtained where k is a constant or a variable depending on v. If