Page:BumsteadContraction.djvu/10

It is scarcely necessary to point out that such problems as we have been considering do not lead to any practicable experimental tests. In order to detect deviations, it would be necessary to measure the periods in question with an accuracy such that the errors should be less than $\scriptstyle {10^{-8}}$, which is quite out of the question at present. This does not however affect the legitimacy of the use of such methods in following out the consequences of the principle; just as the impossibility of actually constructing a reversible engine does not invalidate that method of applying the second law of thermodynamics. More general methods might be used; but there is some advantage, especially in a comparatively new subject, in the simplicity and concreteness of ideas derived from the consideration of special problems.

Applications to Gravitation.

A promising direction in which to look for possible tests of the hypothesis is among the consequences of the deduction that gravitational forces must vary with motion through the ether in the same manner as electrical forces.

If we have a point charge $\scriptstyle {e}$ moving alone the axis of $\scriptstyle {x,}$ with the uniform velocity $\scriptstyle {v,}$ the electric intensity at a point $\scriptstyle {P}$ whose coordinates are $\scriptstyle {r,~\theta }$ is

$\scriptstyle {\mathbf {E} =V^{2}{\frac {e(1-\beta ^{2})}{r^{2}(1-\beta ^{2}\sin ^{2}\theta )^{\frac {3}{2}}}}\mathbf {r} _{1}}$

(1)

where $\scriptstyle {\mathbf {r} _{1}}$ is a unit vector in the direction of $\scriptstyle {r}$ and $\scriptstyle {\beta ={\frac {v}{\text{V}}}}.$^[1] The magnetic force is perpendicular to the plane of $\scriptstyle {\mathbf {r} }$ and $\scriptstyle {\mathbf {v} }$ and has the magnitude

$\scriptstyle {\mathbf {H} ={\frac {v}{{\text{V}}^{2}}}\mathbf {E} \sin \theta ={\frac {ve}{r^{2}}}{\frac {(1-\beta ^{2})\sin \theta }{(1-\beta ^{2}\sin ^{2}\theta )^{\frac {3}{2}}}}}$

(2)

1. Electromagnetic units are used.