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412 we find

or putting

we have

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 ($$\alpha, \beta, \gamma$$) and the modulus $$q$$, the Sun and the planet have a common velocity $$-cq$$ in the direction ($$\alpha, \beta, \gamma$$), 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 ($$\alpha', \beta', \gamma'$$), where—

or

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—