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had already led to the recognition of absence of optical result up to the second order of the ratio of the velocities. Thus the question was suggested whether the above correspondence between the resting and convected systems can be effectively extended up to the second order. It is, in fact, found that the Maxwellian circuital equations of æthereal activity, in the ambient æther, referred to axes moving along with the uniform velocity of convection v. can be reduced to the same form as for axes at rest, up to and including (v/c)², but not (v/c)³, by adopting a local time ε^-1/2(t - vx/c²) as before, but with a new unit ε^-1/2, and also a reduced unit of length parallel to x equal to ε^-1/2, where here and in what follows ε represents 1+v²/c², the units of length along y and z remaining unaltered. It is found that for two æther-fields, one referred to fixed axes and the other to moving axes, standing in this mutual correlation, the electrons, or poles, in approaching which the æthereal electric vector becomes infinite as er^-1, are situated at corresponding points and are of equal values: the relation, exact to the second order, is now that

(f, g, h) and (a, b, c)

in the field belonging to the fixed system of poles correspond to

${\mathsf {E}}^{\frac {1}{2}}\left({\mathsf {E}}^{-{\frac {1}{2}}}f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right)$

and

${\mathsf {E}}^{\frac {1}{2}}\left({\mathsf {E}}^{-{\frac {1}{2}}}a,\ b+4\pi vh,\ c-4\pi vg\right)$

for the field belonging to the convected system; where ε is 1 + v²/c², as above, the factor ε^1/2 being needed to make corresponding poles equal in value instead of merely proportional.

If each pole or electron is connected with a molecule possessing extraneous mass, and it may be having an extraneous field of gravitational and other force of its own, and thereby interacting with other molecules, we shall want to know the forces exerted on that molecule by the surrounding æther, in order to form its own equations of motion, which must be combined with those of the æther-field around it in order to constitute a complete system. But if such other forces are molecularly insignificant, or better, if the electron is a mere passive pole — nucleus of beknottedness in some way — in the æther, conditioned and controlled entirely by the æther around it, just as a vortex ring is conditioned by the fluid in which it subsists and is also carried along thereby, then, as in the familiar hydrodynamics of vortices, the motion of the æther determines the motion of the entirely passive