10.3 There are however other formulae of transformation from the space and time measurements of set to the space and time measurements of set for which Maxwell's equations are invariant. These formulae were discovered first by Larmor for uncharged regions of the field and later by Lorentz for the general case of regions charged or uncharged. Larmor and Lorentz treated their discovery from its formal mathematical side. This aspect of it is important. It enables us, when we thoroughly understand the sequence of events in one electromagnetic field, to deduce innumerable other electromagnetic fields which will be understood equally well. All mathematicians will appreciate what an advance in knowledge this constitutes. But Lorentz also pointed out that if these formulae for transformation could be looked on as the true formulae for transformation from one set to another of the Newtonian group, then all the unsuccessful experiments to detect the earth's motion through the ether could be explained. Namely, the results of the experiments are such as theory would predict. 10.4 The general reason for this conclusion was given by Einstein in a theorem of the highest importance. He proved that the Lorentzian formulae of transformation from one consentient set to another of the Newtonian group—from set to set —are the necessary and sufficient conditions that motion with the one particular velocity (the velocity of light in vacuo) in one of the sets, or , should also appear as motion with the same magnitude in the other set, or . The phenomena of aberration will be preserved owing to the relation between the directions of the velocity expressing the movements in -space and -space respectively. This preservation of the magnitude of a special velocity (however directed) cannot arise with the traditional formulae for relativity. It practically means that waves or other influences advancing with velocity as referred to the space of any consentient set of the Newtonian group will also advance with the same velocity c as referred to the space of any other such set. 10.5 At first sight the two formulae for transformation, namely the traditional formulae and the Lorentzian formulae, appear to be very different. We notice however that, if and be the two consentient sets and if be the velocity of in the -space and of in the -space, the differences between the two formulae all depend upon the square of the ratio of to , where is the velocity of light in vacuo, and are negligible in proportion to the smallness of this number. For ordinary motions, even planetary motions, this ratio is extremely small and its square is smaller still. Accordingly the differences between the two formulae would not be perceptible under ordinary circumstances. In fact the effect of the difference would only be perceived in those experiments, already discussed, whose results have been in entire agreement with the Lorentzian formulae.