Any attempt to replace these formulae by others was to abandon the _role_ of physical explanation and to have recourse to mere mathematical formulae.
But even in physical science difficulties have accumulated round the orthodox formulae.

In the first place Maxwell's equations of the electromagnetic field are not invariant for the transformations of the orthodox formulae; whereas they are invariant for the transformations of the formulae arising from the third of the four cases mentioned above, provided that the velocity c is identified with a famous electromagnetic constant quantity.
Again the null results of the delicate experiments to detect the earth's variations of motion through the ether in its orbital path are explained immediately by the formulae of the third case.

But if we assume the orthodox formulae we have to make a special and arbitrary assumption as to the contraction of matter during motion.

I mean the Fitzgerald-Lorentz assumption.
Lastly Fresnel's coefficient of drag which represents the variation of the velocity of light in a moving medium is explained by the formulae of the third case, and requires another arbitrary assumption if we use the orthodox formulae.
It appears therefore that on the mere basis of physical explanation there are advantages in the formulae of the third case as compared with the orthodox formulae.

But the way is blocked by the ingrained belief that these latter formulae possess a character of necessity.