Page:DeSitterGravitation.djvu/8

Mar. 1911. or

where

Minkowski gives the name "kinetic energy" to the quantity

The equation (12′) thus turns out to be the equation of energy,

If we use Minkowskian velocities and forces, and put

we find similarly from (11′)

The law of the force must be such that the form of the equations of motion is not changed by a Lorentz-transformation. Therefore

must be transformed by the same formulæ as

or, in other words, $$\mathrm{(X), (Y), (Z)}$$ must be linear functions of

the coefficients being invariants of the transformation.

For zero velocities the equations of Newtonian mechanics must be reproduced, therefore the coordinates can only enter by their differences $$x_{1}-x_{2}$$, etc.

Introducing further the condition that the resulting equations must not contain the velocities in the first degree, and certain other simplifications, Poincaré is led to take (l.c., page 174)—