Page:PoincareDynamiqueJuillet.djvu/37

 We have therefore

$$J_{1}=J'_{1}\,$$. C.Q.F.D.

The theorem is thus general, it gives us at the same time a solution of the question we posed at the end of § 1: finding the complementary forces which are unaltered by the transformation. The additional potential (F) satisfies this condition.

So we can generalize the result announced at the end of § 1 and write:

If the inertia of electrons is exclusively of electromagnetic origin, if they are only subject to forces of electromagnetic origin, or to forces generated by the additional potential (F), no experiment can demonstrate absolute motion.

So what are these forces that create the potential (F)? They can obviously be compared to a pressure which would reign inside the electron; all occurs as if each electron were a hollow capacity subjected to a constant internal pressure (volume independent); the work of this pressure would be obviously proportional to the volume changes.

In any case, I must observe that this pressure is negative. Remember the equation (10) of § 6, according to 's hypothesis we write:

$$F=Ar^{3}\theta^{2}$$;

equations (11) of § 6 give us:

$$A=\frac{a}{3b^{4}}.$$

Our pressure is equal to A, with a constant coefficient, which is indeed negative.

Now assessing the mass of the electron – I mean the "experimental mass", that is to say the mass for low velocities – we have (cf. § 6):

$$H=\frac{\varphi\left(\frac{\theta}{k}\right)}{k^{2}r},\quad\theta=k,\quad\varphi=a,\quad\theta r=b;$$

hence

$$H=\frac{a}{bk}=\frac{a}{b}\sqrt{1-V^{2}},$$

I can write for very small V

$$H=\frac{a}{b}\left(1-\frac{V^{2}}{2}\right),$$

so that the mass, both longitudinal and transverse, will be $$\tfrac{a}{b}$$.

Now a is a numerical constant which shows that: the pressure that creates our additional potential is proportional to the 4th power of the experimental mass of the electron.