Perhaps it is this actual velocity.

In this connexion '_in vacuo_' must not mean an absence of events, namely the absence of the all-pervading ether of events.

It must mean the absence of certain types of objects.
In the fourth case, k is positive.

Let us call it h squared, where h will be of the dimensions of a velocity.

This gives a perfectly possible type of transformation formulae, but not one which explains any facts of experience.