co-ordinates; for the term has the opposite sign to the space terms, , , . [Pg 32] Before we analyse further the conditions which define the Lorentz transformation, we shall introduce the light-time, , in place of the time, , in order that the constant shall not enter explicitly into the formulas to be developed later. Then the Lorentz transformation is defined in such a way that, first, it makes the equation a co-variant equation, that is, an equation which is satisfied with respect to every inertial system if it is satisfied in the inertial system to which we refer the two given events (emission and reception of the ray of light). Finally, with Minkowski, we introduce in place of the real time co-ordinate , the imaginary time co-ordinate Then the equation defining the propagation of light, which must be co-variant with respect to the Lorentz transformation, becomes This condition is always satisfied[8] if we satisfy the more general condition that [Pg 33] shall be an invariant with respect to the transformation. This condition is satisfied only by linear transformations, that is, transformations of the type in which the summation over the is to be extended from = 1 to = 4. A glance at equations (23) and (24) shows that the Lorentz transformation so defined is identical with the translational and rotational transformations of the Euclidean geometry, if we disregard the number of dimensions and the relations of reality. We can also conclude that the coefficients must satisfy the conditions Since the ratios of the are real, it follows that all the and the are real, except , , , , , and , which are purely imaginary. [8]That this specialization lies in the nature of the case will be evident later. Special Lorentz Transformation. We obtain the simplest transformations of the type of (24) and (25) if only two of the co-ordinates are to be transformed, and if all the , which determine the new origin, vanish. We obtain then for the indices 1 and 2, on account of the three independent conditions which the relations (25) furnish, [Pg 34] This is a simple rotation in space of the (space) co-ordinate system about -axis. We see that the rotational transformation in space (without the time transformation) which we studied before is contained in the Lorentz transformation as a special case. For the indices 1 and 4 we obtain, in an analogous manner, On account of the relations of reality must be taken as imaginary. To interpret these equations physically, we introduce the real light-time and the velocity of ' relatively to , instead of the imaginary angle . We have, first, Since for the origin of ' i.e., for = 0, we must have , it follows from the first of these equations that and also [Pg 35] so that we obtain