# Page:CarmichealMass.djvu/12

an essential equivalent of $\scriptstyle\left(MVLRC_{1}\right)$; in other words, the experimental demonstration of the formula for transverse mass carries with it the experimental proof of the theory of relativity, provided that postulates $\scriptstyle\left(MVLC_{1}\right)$ are accepted as experimentally proved.

Bucherer[1] has carried out some investigations which have been supposed to furnish this experimental verification for the formula of transverse mass, and hence for the whole theory of relativity. In order to draw this conclusion from Bucherer's direct results it is necessary to make use of a law which we have not yet employed, namely, the law of conservation of electricity which we have stated as postulate $C_2$. Since this law has customarily been accepted and has not yet led to contradictions it should certainly still be supposed to hold. Accepting it, then, we have in Bucherer's results a partial experimental confirmation of the theory of relativity, as we now show.

Bucherer's investigations have to do with the mass of a moving electron. There seems to be no means at hand for a direct measurement of this mass, and Bucherer resorted to the expedient of determining the ratio of charge to mass. Let us denote the charge by e, which we suppose to be constant, in accordance with postulate $C_3$. As before let $m_0$ and $t\left(m_{v}\right)$ denote the mass at rest and the mass when moving with velocity v, of the electron in consideration. Bucherer's experiments were carried out to determine the relation which exists between $e/m_{0}$ and $e/t\left(m_{v}\right)$. The measurements agreed in a remarkable way, not only as to general characteristics but also as to exact numerical results, with the formula[2]

$\frac{e}{t\left(m_{v}\right)}=\frac{e\sqrt{1-\beta^{2}}}{m_{0}}$

Taking this formula as thus experimentally demonstrated we have at once our fundamental relation for transverse mass: $\sqrt{1-\beta^{2}}t\left(m_{v}\right)=m_{0}$.

From this it follows that the experimental demonstration of the theory of relativity is complete when we have proved M, V, L, $C_1$ and $C_3$, provided that one accepts Bucherer's proof of the above relation between $e/m_{0}$ and $e/t\left(m_{v}\right)$. That is, the essentials of the theory of relativity flow from

1. Annalen der Physik (4) 28 (1909), 513-536.
2. As a matter of fact Bucherer did not measure the ratio $e/m_{0}$. Instead of this he considered the ratio $e/t\left(m_{v}\right)$ for a considerable range of values of v and noticed that its value always agreed with the formula $e/t\left(m_{v}\right)=k\sqrt{1-\beta^{2}}$, where k is a constant. It appears natural, then, to assume that $m_{0}=e/k$, whence one has the formula in the text. It should be emphasized that this assumption is necessary in order that the Bucherer results may be associated with our theorem as in the text, and consequently the conclusions there reached can be accepted with no stronger confidence than that which one has in the accuracy of the above assumption. See the next section where a means of experimental verification of the theory of relativity is suggested which does not depend on this assumption for its validity.