three variables x', y', z' define what Newton would call absolute space, and t' corresponds to absolute time.
The second is a system of reference of which the origin and the direction of the axes are chosen according to the needs of the problem in hand. The transformation from x, y, z, t to x', y', z', t', or inversely, is a Lorentz-transformation.
The chief consequence of the principle of relativity is that it is impossible to observe any but relative motions. The "general"` system only differs from any other possible system in that no convention is made as to its origin or the direction of its axes. It is thus not possible, from the point of view of the principle of relativity, to speak of "absolute" velocity or position otherwise than as velocity or position relative to any, not specified, "general " system of reference. If the word "absolute" is used in this sense, it is unobjectionable, but unnecessary. The laws of nature must, of course, be primarily framed with respect to the general system of reference. This does not mean that they assert anything about "absolute” motion or "absolute" time, but only that they must be so built up as to be true in whatever system of reference we choose to use.
We have thus to consider two problems:—
(a) What is the law of force that must replace Newton’s law, and what is the motion of a planet under this law? So far as this differs from ordinary Keplerian motion, we shall have to consider the question whether the differences are large enough to be verified by observation.
(b) A motion being given in any "special" system, how is it expressed in the "general" system, or in any other special system? This only involves the working out of the formulæ of transformation for the required cases.
A third question then arises, viz. whether the "astronomical system," i.e. the coordinates and the time actually used by practical astronomers, coincides or not with any "special" system as defined above. This question is practically not affected by the introduction of the principle of relativity, and we need refer to it only very briefly.
3. I will now begin by stating the necessary formulæ for a Lorentz-transformation in the form which I find most convenient for my purpose. Very probably most of them have already been published elsewhere in the same form, but I found it easier to work them out for myself than to search for them in an unfamiliar literature.
The formulæ become more symmetrical if the unit of time is so chosen that c=1. From an astronomical point of view, however, it is more convenient to retain the ordinary unit of time. Both advantages are combined by using ct as variable instead of t. Derivatives with respect to ct are denoted by Greek letters, thus—