In other words they are the reflections in the space of {alpha} of a set of fixed axes in the space of some other time-system {beta}.

Accordingly the group of dynamical axes required for Newton's Laws of Motion is the outcome of the necessity of referring motion to a body at rest in the space of some one time-system in order to obtain a coherent account of physical properties.

If we do not do so the meaning of the motion of one portion of our physical configuration is different from the meaning of the motion of another portion of the same configuration.

Thus the meaning of motion being what it is, in order to describe the motion of any system of objects without changing the meaning of your terms as you proceed with your description, you are bound to take one of these sets of axes as axes of reference; though you may choose their reflections into the space of any time-system which you wish to adopt.

A definite physical reason is thereby assigned for the peculiar property of the dynamical group of axes.
On the orthodox theory the position of the equations of motion is most ambiguous.