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or, to a first approximation only,

$E={\frac {1}{2}}m_{0}v^{2}$

Therefore the usual formula for kinetic energy in the classical mechanics is only a first approximation.

Since relation (1) is to be true for all velocities v it is obvious that we have

$E_{v}={\frac {m_{0}c^{2}}{\sqrt {1-\beta ^{2}}}}+k,\ E_{0}=m_{0}c^{2}+k,$

where k is a constant, that is, a quantity independent of v. From the first of these equations we conclude further that

$E_{v}=c^{2}\cdot t\left(m_{v}\right)+k$

so that the total energy of a body, decreased by the constant k, is directly proportional to its transverse mass. In case the body is at rest its mass in one direction is the same as in another; hence $m_{0}=t\left(m_{0}\right)$. Bearing this in mind we have the following theorem:

Theorem VII. Let $m_{0}$ be the mass of a body when at rest with respect to a given system of reference and let $t\left(m_{v}\right)$ denote its transverse mass when it is moving with the velocity v (the case v = 0 is not excluded). Then the total energy $E_{v}$ which it possesses is $c^{2}t\left(m_{v}\right)+k$, where k is a constant.

The following relations are immediate consequences of equations written out above:

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

Now, suppose that an experimenter contributes to a body which is at rest a known amount of energy and determines the velocity which this causes the body to acquire. If the two measurements are made with sufficient accuracy one will be able, by substituting the results in the first of the above equations, to determine in this way the velocity of light. Actually to carry out this remarkable method for measuring c would doubtless be very difficult; but the obvious great importance of the result is certainly such as to justify a careful consideration of the problem. If the value of c determined in this way should agree well with its value as otherwise found, this would give us an interesting confirmation of the theory of relativity.

Let us consider the mass of a rotating top, the mass being measured along the axis of rotation. According to our results this mass should be different from that of the same top when at rest, and the difference should bear a definite relation to the amount of energy which is involved