# Page:CarmichealMass.djvu/17

in the rotation. If the measurements here involved could be made with sufficient accuracy we would have another means, independent of light itself, for the measurement of the light-velocity c. Again, this experiment would afford us a measure of transverse mass and in that way could lead to a confirmation of the theory of relativity, provided that we assume c as known from independent measurements; and this confirmation, it is to be noticed, would be independent of electrical considerations.

It seems to be impossible to determine the constant k which enters into the above discussion. But in the absence of any evidence to the contrary it would appear natural tentatively to assume that k is zero. On the basis of this assumption we should have the following remarkable conclusions: The mass of a body at rest is simply the measure of its internal energy. The transverse mass of a body in motion is the measure of its internal energy and its kinetic energy taken together.[1] Its longitudinal mass is its total energy multiplied by a simple factor. (The longitudinal mass, therefore, bears a simple ratio to the total energy.) One can hardly resist the conclusion that the transverse mass of a body depends entirely on its energy, and therefore that matter is merely one manifestation of energy.

## § 8. Remarks on the Principle of Least Action.

In the preceding section we have seen that in the theory of relativity the classical formula $E=\tfrac{1}{2}mv^{2}$ for the measure of kinetic energy is true only as a first approximation. This is due to the fact that mass is a variable quantity. But the conclusion does not appear to necessitate our surrender of the law of conservation of energy.

The same causes which lead to a modification in the formula for E will also require a corresponding modification in the value of the action A as defined in § I. The question arises as to whether the principle of least action is left intact. I cannot enter upon the investigation here; but the problem seems to me to be of importance, and consequently I am stating it in the hope that some one will be led to consider its solution.

Undoubtedly the principle of least action is one which should be given up only when there are strong reasons for it. It is a mathematical formulation of the law that nature accomplishes her ends with the least possible expenditure of labor, so to speak. Certainly this law is one which appeals to our minds with strong force. There is something about it which is aesthetically satisfying in a high degree. It seems to me, however, that a fresh study of it should be made in the light of the theory of relativity.

1. Compare Lewis and Tolman, Phil. Mag. (6) 18 (1909), p. 521.