Page:The Meaning of Relativity - Albert Einstein (1922).djvu/70

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58
THE MEANING OF RELATIVITY

must hold. We shall now show that this equation leads to the same law of motion of a material particle as that already obtained. Let us imagine the matter to be of infinitely small extent in space, that is, a four-dimensional thread; then by integration over the whole thread with respect to the space co-ordinates , we obtain

Now is an invariant, as is, therefore, also . We shall calculate this integral, first with respect to the inertial system which we have chosen, and second, with respect to a system relatively to which the matter has the velocity zero. The integration is to be extended over a filament of the thread for which may be regarded as constant over the whole section. If the space volumes of the filament referred to the two systems are and respectively, then we have

and therefore also

If we substitute the right-hand side for the left-hand side in the former integral, and put outside the sign