Page:Deux Mémoires de Henri Poincaré.djvu/5

 at the moment t is equal to that which is in $$d\tau'$$ at the moment $$t'$$. Consequently, if ρ and $$\rho'$$ are the densities of these charges,

and, by virtue of (5)

By this formula, combined with (4), we deduce again

These are the transformation formulas for the convection current.

For other physical quantities such as electric and magnetic forces, it is necessary to follow a less direct method; we will seek, perhaps with a little groping, the formulas of transformation suitable to ensure the invariance of the electromagnetic equations.

The formulas (4) and (7) are not in my memoir of 1904. Because I had not thought of the direct way which led there, and because I had the idea that there is an essential difference between systems x, y, z, t and $$x', y', z', t'$$. In one we use - such was my thought - coordinate axes which have a fixed position in the aether and which we can call "true" time; in the other system, on the contrary, we would deal with simple auxiliary quantities whose introduction is only a mathematical artifice. In particular, the variable $$t'$$ could not be called "time" in the same way as the variable t.

In this order of ideas I did not think of describing the phenomena in the system $$x', y', z', t',$$ exactly in the same way as in system x, y, z, t, and I did not define by the equations (3) and (7) the quantities ξ', η', ζ', ρ' which will correspond to ξ, η, ζ, ρ. It is rather by groping that I arrived at my formulas of transformation which, with our current notation, take the form

and that I wanted to choose, so as to obtain in the new system the simplest equations. Later, I could see in the paper of Poincaré that when proceeding more systematically I could have reached an even greater