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 $$C_1$$. The sum total of momentum in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without.

$$C_2$$. The sum total of energy in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without.

$$C_3$$. The sum total of electricity in any isolated system remains unaltered, whatever changes may take place in the system, provided that the system as a whole neither receives electricity from nor gives out electricity to bodies not belonging to the system.

The "action" of a moving body in passing from one position to another may be defined as the space integral of the momentum taken over the path of motion. If we denote this action by A we have therefore

Now ds = vdt, so that we have also

If several bodies are involved we have

where the summation is for the various bodies in the system.

We may state the fundamental principle of least action in the following form:

. The free motion of a conservative system between any two given configurations has the property that the action A is a minimum, the admissible values $$\overline{A}$$ of the action with which A is compared being obtained from varied motions in which the total energy has the same constant value as in the actual free motion.

§ 2. .
Suppose that we have two systems of reference $$S_1$$ and $$S_2$$ moving with a relative velocity v. We inquire as to whether, and in what way, the mass of a body as measured on the two systems depends on v. Will a given body have the same measure of mass when that mass is estimated in units of $$S_1$$ and in units of $$S_2$$? And will the mass of a body depend on the direction of its motion by means of which that mass is measured? Our purpose in this section is to answer these two questions.

The two most important directions in which to measure the mass of a body are, first, that perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$ and, secondly, that parallel to this line of motion. For convenience