Page:DeSitterGravitation.djvu/26

or

$n't'={\frac {\pi }{2}}-\epsilon ,\qquad {\frac {3\pi }{2}}+\epsilon ,\qquad {\frac {5\pi }{2}}-\epsilon ,\qquad {\text{ etc.}}$

The intervals are therefore—

${\frac {\pi +2\epsilon }{n'}},\qquad {\frac {\pi -2\epsilon }{n'}},\qquad {\text{ etc.}}$

or

${\frac {\pi }{n'}}+{\frac {2a\beta q}{c{\sqrt {1-q^{2}}}}},\qquad {\frac {\pi }{n'}}-{\frac {2a\beta q}{c{\sqrt {1-q^{2}}}}},\qquad {\text{ etc.}}$

Let the aberration-time be called $b/c$. We find easily—

$b^{2}=\left(a\mathrm {K} \pm \beta 'qb\right)^{2}+q^{2}b^{2}\left(1-\beta '^{2}\right),$

the upper sign being taken for the crossings through the positive axis of $y$ (first and third epoch), and the lower sign for those in which $y$ is negative (middle epoch). We find then—

$b_{2}-b_{1}=b_{2}-b_{3}=-{\frac {2\beta 'a\mathrm {K} q}{1-q^{2}}}=-{\frac {2a\beta q}{\sqrt {1-q^{2}}}}$

The corrections to be applied to the true intervals to derive the observed intervals are—

${\frac {b_{2}-b_{1}}{c}},\qquad {\text{ and}}\qquad {\frac {b_{3}-b_{2}}{c}}.$

The observed intervals are thus seen to be equal, as in the system ($x,y,z,t$).

Dr. C. V. Burton,^[1] following an idea of Maxwell, has proposed to determine the velocity of the solar system "with respect to the aether" from observations of eclipses of Jupiter’s satellites. Apart from the complications introduced by the excentricities and inclinations, and by the observer being on the Earth instead of on the Sun, his method is to determine $\beta q$ from the observed inequality in the intervals. If the principle of relativity be true, the result must necessarily be nil. The method could thus be used to verify the principle of relativity.

Mathematically speaking, the dynamical aspect of the principle is not directly concerned in this verification. The observable effect would be nil whatever the motions of the observer and the signaller, which are subjected to the same Lorentz-transformation, may be, even if they were dynamically impossible, so long as the principle is true for the geometrical relations of time, space, and

1. Phil. Mag., 1910 March, page 417.