Page:TolmanNon2.djvu/2

it has been believed by Professor Lewis and the writer, that in general, without respect to direction, the expression $m_{0}/{\sqrt {1-u^{2}/c^{2}}}$ is best suited for the mass of a moving body. They have already shown (loc. cit.), from the theory of relativity and the principles of non-Newtonian mechanics outlined above, that the consideration of a “transverse collision” between two moving bodies does lead to this expression for the mass of a moving body: and the purpose of the present article is to show that the consideration of any type of collision also leads to the same expression.

The immediate occasion of the present article is a recent attempt made by Mr. Norman Campbell^[1] to show that the consideration of a “longitudinal collision” does not lead to the expression $m_{0}/{\sqrt {1-u^{2}/c^{2}}}$ for the mass of a moving body. There appears, however, to be an obvious error in his reasoning. Mr. Campbell wishes to find a relation between the mass of a body and its velocity and yet assumes that the mass of each of his bodies is the same after collision as before, although the velocities of course have changed (see equation (A) p. 627). Thus, although endeavouring to determine how the mass of a body depends on the velocity, he assumes in formulating his fundamental equation that it does not depend on the velocity at all^[2].

Longitudinal Collision.

Consider a system of Cartesian coordinates and two bodies moving in the X direction with the velocities +u and -u in such a way that a "longitudinal collision" will take place. Suppose the bodies are elastic and perfectly similar, each having the mass $m_{0}$ when at rest. On collision the bodies will evidently come gradually to rest, and then under the action of the elastic forces developed start up and move back on their original paths with the respective velocities -u and +u of the same magnitude as before.

Let us now consider how the collision will appear to an observer who is moving past the above system of coordinates with the velocity $v$ in the X direction. Let $u_{1}$ and $u_{2}$ be the velocities of the two bodies as they appear before collision to this new observer. From Einstein’s formulæ for the

1. Phil. Mag. xxi. p. 626 (1911).
2. In the same article, Mr. Campbell has also criticised the writer for referring the acceleration of a hotly under consideration to moving axes which have at the moment in question the same velocity as the body itself. As this is a procedure which has long been familiar to students of theoretical mechanics, has not in the past led to erroneous results, and in the cases under consideration leads to self-consistent conclusions, the writer cannot agree with Mr. Campbell's criticism.