The equations (6) and (7_b_) determine the constants _a_ and _b_. By inserting the values of these constants in (5), we obtain the first and the fourth of the equations given in Section XI. image047 Thus we have obtained the Lorentz transformation for events on the _x_-axis. It satisfies the condition _x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (8a). The extension of this result, to include events which take place outside the _x_-axis, is obtained by retaining equations (8) and supplementing them by the relations image048 In this way we satisfy the postulate of the constancy of the velocity of light _in vacuo_ for rays of light of arbitrary direction, both for the system _K_ and for the system _K′_. This may be shown in the following manner. We suppose a light-signal sent out from the origin of _K_ at the time _t_ = 0. It will be propagated according to the equation image049 or, if we square this equation, according to the equation _x_2 + _y_2 + _z_2 – _c_2_t_2 = 0 . . . . . (10). It is required by the law of propagation of light, in conjunction with the postulate of relativity, that the transmission of the signal in question should take place—as judged from _K′_—in accordance with the corresponding formula _r′_ = _ct′_ or, _x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_). In order that equation (10_a_) may be a consequence of equation (10), we must have _x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2) (11). Since equation (8_a_) must hold for points on the _x_-axis, we thus have σ = 1. It is easily seen that the Lorentz transformation really satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_) and (9), and hence also of (8) and (9). We have thus derived the Lorentz transformation. The Lorentz transformation represented by (8) and (9) still requires to be generalised. Obviously it is immaterial whether the axes of _K′_ be chosen so that they are spatially parallel to those of _K_. It is also not essential that the velocity of translation of _K′_ with respect to _K_ should be in the direction of the _x_-axis. A simple consideration shows that we are able to construct the Lorentz transformation in this general sense from two kinds of transformations, viz. from Lorentz transformations in the special sense and from purely spatial transformations. which corresponds to the replacement of the rectangular co-ordinate system by a new system with its axes pointing in other directions. Mathematically, we can characterise the generalised Lorentz transformation thus: It expresses _x′, y′, x′, t′_, in terms of linear homogeneous functions of _x, y, x, t_, of such a kind that the relation _x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2 (11_a_). is satisficd identically. That is to say: If we substitute their expressions in _x, y, x, t_, in place of _x′, y′, x′, t′_, on the left-hand side, then the left-hand side of (11_a_) agrees with the right-hand side.