Technically, the whole of the special theory is contained in the Lorentz transformation. This transformation has the advantage that it makes the velocity of light the same with respect to any two bodies which are moving uniformly relatively to each other, and, more generally, that it makes the laws of electromagnetic phenomena (Maxwell's equations) the same with respect to any two such bodies. It was for the sake of this advantage that it was originally introduced; but it was afterwards found to have wider bearings and a more general justification. In fact, it may be said that, given sufficient logical acumen, it could have been discovered at any time after it was known that light is not propagated instantaneously. It has grown by this time very familiar—so familiar that I have even seen it quoted (quite correctly) in an advertisement of Fortnum and Mason's. Nevertheless, it is, I suppose, desirable to set it forth. In its simplest form it is as follows: Suppose two bodies, one of which () is moving relatively to the other () with velocity v parallel to the -axis. Suppose that an observer on observes an event which he judges to have taken place at time , by his clocks, and in the place whose co-ordinates, for him, are , , . (Each observer takes himself as origin.) Suppose that an observer on judges that the[Pg 50] event occurs at time and that its co-ordinates are , , . We suppose that at the time when the two observers are at the same place, and also . It would formerly have seemed axiomatic that we should have . Both observers are supposed to employ faultless chronometers, and, of course, to allow for the velocity of light in estimating the time when the event occurs. It would be thought, therefore, that they would arrive at the same estimate as to the time of the occurrence. It would also have been thought that we should have: Neither of these, however, is correct. To obtain the correct transformation, put: where is, as always, the velocity of light. Then: For the other co-ordinates , , we still have, as before: It is the formulæ for and that are peculiar. These formulæ contain, implicitly, the whole of the special theory of relativity. The formula for embodies the FitzGerald contraction. Lengths on either body, as estimated by an observer on the other, will be shorter than as estimated by an observer on the body on which the lengths are: the longer length will have to the shorter the ratio . More interesting, however, is the effect as regards time. Suppose that an observer on the body judges two events at and to be simultaneous, and both at time . Then an observer on will judge that they occur at times , where: [Pg 51] and therefore: