The directions of motion in the space of {alpha} due to rest in a point of {beta} is called the '{beta}-direction in {alpha}' and the direction of motion in the space of {beta} due to rest in a point of {alpha} is called the '{alpha}-direction in {beta}.' Consider a motion in the space of {alpha} consisting of a certain velocity in the {beta}-direction of {alpha} and a certain velocity at right-angles to it.

This motion represents rest in the space of another time-system--call it {pi}.

Rest in {pi} will also be represented in the space of {beta} by a certain velocity in the {alpha}-direction in {beta} and a certain velocity at right-angles to this {alpha}-direction.

Thus a certain motion in the space of {alpha} is correlated to a certain motion in the space of {beta}, as both representing the same fact which can also be represented by rest in {pi}.

Now another time-system, which I will name {sigma}, can be found which is such that rest in its space is represented by the same magnitudes of velocities along and perpendicular to the {alpha}-direction in {beta} as those velocities in {alpha}, along and perpendicular to the {beta}-direction, which represent rest in {pi}.
The required axiom of kinetic symmetry is that rest in {sigma} will be represented in {alpha} by the same velocities along and perpendicular to the {beta}-direction in {alpha} as those velocities in {beta} along and perpendicular to the {alpha}-direction which represent rest in {pi}.
A particular case of this axiom is that relative velocities are equal and opposite.