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that these two changes of momentum are equal. Hence to A it appears th at the mass of the ball $B_{1}$ is smaller than that of the ball $B_{2}$ in the ratio ${\sqrt {1-\beta ^{2}}}:1$.

Similarly, it may be shown that to C it appears that the mass of the ball $B_{2}$ is smaller than that of $B_{1}$ in the ratio ${\sqrt {1-\beta ^{2}}}:1$.

From the general theorems concerning the measurement of length (in the theory of relativity) it follows that if the ball which has been constructed by A were transferred to $C$'s system it would be impossible for C to distinguish $A$'s ball from his own by any considerations of shape and size. Likewise, as A looks at them from his own system he is similarly unable to distinguish them. It is therefore natural to take the mass of $C$'s ball as that which $A$'s would have if it had the velocity v with respect to $S_{1}$ of the system $S_{2}$. Thus we obtain a relation existing between the mass of a body in motion and at rest.

Now, "mass" as we have measured it above is the transverse mass of our definition. From the argument just carried out we are forced to conclude that the transverse mass of a body in motion depends (in a certain definite way) on the velocity of that motion. The result may be formulated as follows:

Theorem I. Let $m_{0}$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by $t\left(m_{v}\right)$ its transverse mass, that is, its mass in a direction perpendicular to its line of motion. Then we have

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

where $\beta =v/c$ and c is the velocity of light $\scriptstyle (MVLRC_{1})$.^[1]

In the statement of this theorem we have tacitly assumed that the mass of a body at rest relative to S, when measured by means of units belonging to S, is independent of the direction in which it is measured. If this assumption were not true we should have a means of detecting the motion of S, a conclusion which is in contradiction to postulate M.

In order to find the longitudinal mass of a moving body we first find the relation which exists between longitudinal mass and transverse mass. We employ for this purpose the elegant method of Bumstead.^[2]

Let us as usual consider two systems of reference $S_{1}$ and $S_{2}$ moving with a relative velocity v, observers A and B being stationed on $S_{1}$ and $S_{2}$ respectively. Suppose that B performs the following experiment:

1. These letters attached to the theorem indicate that in its proof we have employed postulates M, V, L, R, $C_{1}$. Compare a similar usage in my previous paper.
2. American Journal of Science (4) 26 (1908), pp. 498-500.