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Mar. 1911. system. The problem has been exhaustively treated by Anding (Encyclopädie der Math. Wissensch., vi., 2, page 3) and Seeliger ("Ueber die sogenannte absolute Bewegung," Münchener Sitzungsber., 1906, page 85) and others. Since the nodes are fixed under the new law, and also the perihelia are practically fixed, with the exception of Mercury (for which see art. 13), and the correction (42) to the constant of precession is insensible, the problem is not changed in aspect by the introduction of the principle of relativity, and we need not further refer to it here.

18. Astronomical time is defined by the rotation of the Earth, and is measured by the hour-angle of the axis of $$x$$. Such deviations from uniformity as are derived from the action of known causes—which are, however, negligible, and remain negligible under the new law—are allowed for, and the thus corrected rotation is supposed to be uniform. Now a motion which is in a system ($$x, y, z, t$$) described by the formulæ of uniform rotation, is no longer so described in another system ($$x', y', z', t'$$). The definition of astronomical time would thus be dependent on the system of reference.

But the axis of $$x$$ cannot be observed. Time is determined by the observation of the hour-angles of stars. In the new system ($$x', y', z', t'$$) the direction in which we see the stars is also changed, and a compensation is effected.

Determinations of time are made at intervals, and are used to regulate the clock. Equal intervals on the clock are then supposed to be equal intervals of "time." The construction of the clock depends on the laws of the simple pendulum, which swings under the action of gravitation. We would thus be led to identify astronomical time with the proper-time of the pendulum-bob. Instead of this we can, of course, take the proper-time of the transit-room, or of the centre of the Earth. In this time as variable the motion of the Earth, adopting the law I. and neglecting its mass, is strictly Keplerian. But if we wish to consider the whole solar system, it is better to introduce heliocentric (or "barocentric") time. The difference has been found to be insensible [equation (34)]. The whole question of the definition of astronomical time is thus only of academic interest. So far as the interpretation of observations is concerned, we can identify astronomical time with the variable $$t$$ of any system of reference we wish to use.