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 Suppose now that the law of some phenomenon as observed on $$S'$$ is given by the equation

and we desire to know the expression of this law on S. We substitute for x', y', z', t', their values in terms of x, y, z, t given above; and thus obtain an equation stating the law in question.

From these considerations it appears that in many of our problems, namely, in those which have to do at once with two or more systems of reference, the time and space variables taken together play the rôle of four variables each having to do with one dimension of a four-dimensional continuum.

This conclusion alone raises philosophical questions of profound importance concerning the nature of space and time; but into these we cannot enter here.

VIII. A Maximum Velocity for Material Bodies.
There are several ways by which it may be shown that a material body cannot have a velocity as great as that of light. One of the simplest is that which comes from a consideration of mass. Let us consider the equation

where $$m_0$$ is the mass of a body at rest relative to a given system of reference S, $$l_v$$ is the longitudinal mass of the body moving with a velocity v with respect to S. If we consider larger and larger values of the velocity v we see that $$l_v$$ increases and becomes infinite as v approaches c. This is equivalent to saying that the longitudinal mass of any material body becomes infinite as the velocity of that body approaches c. Therefore it would require an infinite force to give to a material body the velocity c; that is, c is a maximum velocity which the velocity of a material body may approach but can never reach.

We may obtain the same result in a different way, and thus arrive at a deeper understanding of the matter. From the first formula of transformation given in the preceding section we have

Now suppose that v/c > 1 and that t and x are real. Then $$t'$$ is imaginary. Hence, if two systems have a relative velocity greater than that of light, the time measurement in one of them is expressed as an imaginary