# Page:DeSitterGravitation.djvu/25

we find

 $n't'=nt-\frac{an\beta q}{c}\sin\ nt,\quad n'=n\sqrt{1-q^{2}},$

or putting

 $\epsilon=\frac{an\beta q}{c},$

we have

 $\phi-\epsilon\ \sin\phi=n't',$ $x'_{1}=a\ \cos\ \phi,\qquad y'_{1}=aK\ \sin\phi.$ (43)

If the orbit in (x, y, z, t) is a "perturbed" ellipse, the formulæ of transformation are more complicated, but the differences between the real motion and pure elliptic motion remain of the second order and periodic in terms of t', if they were so in terms of t.

16. One of the consequences of the principle of relativity is that it must be impossible by observations on bodies belonging to one and the same system to detect a motion of the whole system. Suppose in the system of reference (x, y, z, t) the Sun to be at rest, and a planet to describe a circle with uniform velocity. This, of course, is a dynamically possible state of motion under the new law as well as under the old. In the system (x', y', z', t') derived from (x, y, z, t) by a Lorentz-transformation with the axis (α, β, γ) and the modulus q, the Sun and the planet have a common velocity -cg in the direction (α, β, γ), and the relative orbit is no longer circular, but is defined by (43). Let, again, the orbital plane be chosen as plane of (x, y) in the first system of reference, and let the axis of x be perpendicular to the axis of the transformation. The observer belonging to the system has no means of ascertaining the position of the plane of (x', y'), he can only observe the plane of the orbit, i.e. the plane of $\left(x'_{1},\ y'_{1}\right)$. To him the velocity of the Sun is thus in the direction (α', β', γ'), where—

 $\alpha'=0,\qquad\gamma'=\sqrt{1-\beta'^{2}}$ $\beta'=\cos(A+I),\ if\ \beta=\cos\ A,$

or

 $\beta'K=\beta\sqrt{1-q^{2}}$

Let the observer be on the Sun, and let a signal be sent him, from the planet every time when this latter crosses the axis of y.

In the system (x, y, z, t) the intervals between these crossings are equal, and also the times required by the signal to reach the observer are equal: he will observe signals at equal intervals.

In the system (x’, y', z', t') the intervals between the crossings are unequal, and the aberration-times are unequal, and these two effects must cancel each other.

The times of crossing are—

 $\phi=\frac{\pi}{2},\qquad\frac{3\pi}{2},\qquad\frac{5\pi}{2}\qquad etc.$