Page:Lorentz Grav1900.djvu/11

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On the other hand ${\mathfrak {v}}'={\mathfrak {v}}$ ; hence, by (I) and (II),

${\mathfrak {d}}'=-{\mathfrak {d}}$ and ${\mathfrak {H}}'=-{\mathfrak {H}}$.

Now let us put in the field, produced by the pair of ious, a similar pair with charges e and $e'=-e$, and moving with the common velocity ${\mathfrak {v}}$. Then, by (10) — (13),

$k_{2}=-{\frac {\beta }{\alpha }}k_{1},\ k_{3}=-{\frac {\beta }{\alpha }}k_{1},\ k_{4}=k_{1}.$

The total force on the positive particle will be

$k_{1}+k_{2}=k_{1}\left(1-{\frac {\beta }{\alpha }}\right)$

and that on the negative ion

$k_{3}+k_{4}=k_{1}\left(1-{\frac {\beta }{\alpha }}\right)$.

These forces being equal and having the same direction, there is no force tending to separate the two ions, as would be the case in an electric field. Nevertheless, the pair is acted on by a resultant force

$2k_{1}\left(1-{\frac {\beta }{\alpha }}\right)$.

If now β be somewhat larger than $\alpha$, the factor $2\left(1-{\frac {\beta }{\alpha }}\right)$ have a certain negative value –ε, and our result may be expressed as follows:

If we wish to determine the action between two ponderable bodies, we may first consider the forces existing between the positive ions in the one and the positive ions in the other. We then have to reverse the direction of these forces, and to multiply them by the factor ε. Of course, we are led in this way to Newton's law of gravitation.

The assumption that all ponderable matter is composed of positive and negative ions is no essential part of the above theory. We might have confined ourselves to the supposition that the state of the aether which is the cause of gravitation is propagated in a similar way as that which exists in the electromagnetic field.

Proceedings Royal Acad. Amsterdam. Vol. II.