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Mar. 1911. where $$p$$ is the constant of general precession in longitude. We have thus—

If we take for the axis of the transformation the direction of the Sun’s motion relatively to the fixed stars, and for $$q$$ the velocity of this motion, divided by the velocity of light, so that in the system ($$x', y', z', t')$$ the mean velocity of the stars is zero, we have—

We find—

This is, on the basis of the principle of relativity, the theoretical difference between the constant of procession as determined from the fixed stars (system $$x', y', z', t'$$) and from the motions in the solar system (system $$x, y, z, t$$).

15. If we take the orbital plane in the system ($$x, y, z, t$$) as the plane of ($$x, y$$), and choose the axis of $$x$$ perpendicular to the axis of the transformation, we have—

and the formulæ for transformation to simultaneous relative coordinates in the system ($$x', y', z', t'$$) become—

The problem is now, if $$x, y$$ are given as functions of $$t$$, to find the expression of $$x',y',z'$$ as functions of $$t'$$. Knowing that the orbit remains plane, we can eliminate $$z'$$ by combining with our Lorentz-transformation a rotation round the axis of $$x'$$ by an angle given by—

from which

The new plane of $$x'_{1},\ y'_{1}$$ is thus the transformed orbital plane, and we have—

If, to take a simple example, the orbit in ($$x, y, z, t$$) were a circle—