Hendrik Lorentz

“ If, finally, we suppose these forces to be limited to such small distances that, for particles acting on each other, the difference of local times may be neglected, one of the particles, together with those which lie in its sphere of attraction or repulsion, will form a system which undergoes the often mentioned deformation. In virtue of the second hypothesis of § 8 we may therefore apply to the resulting molecular force acting on a particle, the equation (21) . ”

“ In other terms, whatever be the nature of the particles composing a ponderable body, so long as they do not move relatively to each other, we shall have between the forces acting in a system (Σ' ) without, and the same system (Σ) with a translation, the relation specified in (21) , if, as regards the relative position of the particles, Σ' is got from Σ by the deformation (kl, l, l) , or Σ from Σ' by the deformation . ”

“ In the second place Trouton and Noble [3] have endeavoured to detect a turning couple acting on a charged condenser, whose plates make a certain angle with the direction of translation. The theory of electrons, unless it be modified by some new hypothesis, would undoubtedly require the existence of such a couple. In order to see this, it will suffice to consider a condenser with aether as dielectricum. It may be shown that in every electrostatic system, moving with a velocity , [4] there is a certain amount of electromagnetic momentum. ”

“ It appears from the second of the formulae (30) that my theory leads likewise to an equation of the form (35) ; only Abraham's function ψ (β) must be replaced by
. Hence, my theory requires that, if we substitute this value for ψ (β) in (34) , these equations shall still hold. Of course, in seeking to obtain a good agreement, we shall be justified in giving to k1 and k2 other values than those of Kaufmann, and in taking for every measurement a proper value of the velocity w, or of the ratio β. ”

“ Strictly speaking, the formula (28) may only be applied in the case of a uniform rectilinear translation. On account of this circumstance — though (29) is always true — the theory of rapidly varying motions of an electron becomes very complicated, the more so, because the hypothesis of § 8 would imply that the direction and amount of the deformation are continually changing. It is even hardly probable that the form of the electron will be determined solely by the velocity existing at the moment considered. ”

“ As to the experiments of TROUTON and NOBLE, their negative result becomes at once clear, if we admit the hypotheses of §8. It may be inferred from these and from our last assumption (§ 10) that the only effect of the translation must have been a contraction of the whole system of electrons and other particles constituting the charged condenser and the beam and thread of the torsion-balance. Such a contraction does not give rise to a sensible change of direction. ”

“ It need hardly be said that the present theory is put forward with all due reserve. Though it seems to me that it can account for all well established facts, it leads to some consequences that cannot as yet be put to the test of experiment. ”

“ In doing so, we shall give the symbol , which, in (17) , relates to the instant at which local time in dS is , a slightly different meaning. Distinguishing by the value of r' for the centre A, we shall understand by the value of the density existing in the element dS at the point (x, y, z) , at the instant t0 at which the local time of A is .
It may be seen from (5) that this instant precedes that for which we have to take the numerator in (17) by
units of time. ”

“ It will suffice for our purpose to consider two special cases. The first is that of an electrostatic system, i.e. a system having no other motion but the translation with the velocity w. In this case , and therefore, by (12) , . Also, φ' is independent of t' , so that the equations (11) , (13) and (14) reduce to
After having determined the vector b' by means of these equations, we know also the electric force acting on electrons that belong to the system. ”