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rays of light. But in practice the coordinates are always derived from the integration of equations of motion, and they could not be transformed by a Lorentz-transformation unless the right-hand members of these equations of motion satisfy the conditions imposed by the principle of relativity.

If we take for $cq$ the Sun’s velocity relative to the fixed stars, the effect to be expected, in case the principle of relativity were not true, would be equivalent to an inequality in the time of eclipse of $0^{\mathrm {s} }.10$ with a period of 12 years, corresponding to

$0^{\circ }.00023,\quad 0^{\circ }.00012,\quad 0^{\circ }.00006,\quad 0^{\circ }.00002,$

in the longitudes of the four satellites. Our observations, as well as our knowledge of the principal constants of the theory, must become many times more accurate than they are now, before it will be possible to detect the existence of an inequality of this amount.

17. The velocity of the solar system relatively to the fixed stars, on the other hand, is the same in any system of reference. If in the system ($x',y',z',t'$) in which the Sun has the velocity $-cq$, the star has no velocity ($\xi '_{\star }=\eta '_{\star }=\xi '_{\star }=0$), then, transforming back to the system ($x,y,z,t$), in which the Sun has no velocity, we find for the star the velocity $\xi _{\star }=\alpha cq,\ \eta _{\star }=\beta cq,\ \zeta _{\star }=\gamma cq,$ (neglecting terms of the second order).

If in the system ($z',y',z',t'$) two groups of stars have any systematic motion relatively to each other, or all stars have a systematic motion relatively to the axes of reference, the same will be the case in the system ($x,y,z,t$), apart from small changes (of the second order) in the constants defining the direction of motion relatively to the ecliptic, which are, however, of no importance, as they are constant corrections to quantities whose values must be derived from observation. If no such systematic motion exists in ($x',y',z',t'$), neither does it exist in ($x,y,z,t$).

This brings us to the question of the astronomical system of coordinates. Let this be ($\mathrm {x,y,z,t}$). The observations are referred to a system of axes $\mathrm {x',y',z'}$, which is the system of the fundamental catalogue. This system is reduced to ($\mathrm {x,y,z}$) by a rotation—

${\begin{aligned}&\mathrm {x} =\mathrm {x} '\ \cos \ pt+\mathrm {y} '\ \sin \ pt,\\&\mathrm {y} =\mathrm {y} '\ \cos \ pt+\mathrm {x} '\ \sin \ pt,\\&\mathrm {z} =\mathrm {z} ',\end{aligned}}$

where $p$ is the constant of precession. This constant is so determined that the fixed stars shall have no systematic rotation with respect to the system ($\mathrm {x,y,z}$), and the assumption is then made that ($\mathrm {x,y,z}$) coincides with ($x,y,z$), the system to which the equations of motion are referred. Whether this is so or not can only be decided by comparison with the motions in the solar