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in distinguishing these we shall speak of the "transverse mass" of a body as that with which we have to deal when we are concerned with the motion of the body in a direction perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$; when the motion is parallel to this line we shall speak of the "longitudinal mass" of the body.

Lewis and Tolman^[1] determine what they call the "mass of a body in motion," employing for this purpose a very simple and elegant method. This "mass" is what we have just defined as the transverse mass of the body. We employ the excellent method of these authors in deriving the formula for transverse mass.

Suppose that an experimenter A on the system $S_{1}$ constructs a ball $B_{1}$ of some rigid elastic material, with unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$, the units of measurement employed being those belonging to $S_{1}$. Likewise suppose that an experimenter C on $S_{2}$ constructs a ball $B_{2}$ of the same material, also of unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$; we suppose that the measurements made by C are with units belonging to $S_{2}$. Assume that the experiment has been so planned that the balls will collide and rebound, over their original paths, the path of each ball being thought of as relative to the system to which it belongs.

Now the relation of the ball $B_{2}$ to the system $S_{1}$ is the same as that of the ball $B_{1}$ to the system $S_{2}$, on account of the perfect symmetry which exists between the two systems of reference in accordance with the results of the previous paper (already referred to). Therefore the change of velocity of $B_{2}$ relative to its starting point on $S_{2}$ as measured by A is equal to the change of velocity of $B_{1}$ relative to its starting point on $S_{1}$ as measured by C. Now velocity is equal to the ratio of distance to time: and in the direction perpendicular to the line of relative motion of the two systems the units of length are equal; but the units of time are unequal. Hence to either of the observers the change of velocity in the two balls, each with respect to its starting point on its own system, will appear to be unequal.

To A the time unit on $S_{2}$ appears to be longer than his own in the ratio ${\sqrt {1-\beta ^{2}}}:1$ (see previous paper, theorem IV., p. 167). Hence to A it must appear that the change in velocity of $B_{2}$ relative to its starting point is smaller than that of $B_{1}$ relative to its starting point in the ratio ${\sqrt {1-\beta ^{2}}}:1$. But the change in velocity of each ball multiplied by its mass gives its change in momentum. From postulate $C_{1}$ it follows

1. Phil. Mag., 18 (1909), 510-523. See especially pp. 517-518.