U_{1} = β(_u_{1} - _v_)/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, and T = P_x_ + Q_y_ + R_z_ + S_t_. But we assumed that OO′, _i.e._, O_x_, is an axis of symmetry. It follows from this assumption that Q = R = 0. We thus obtain the simplified results T = P_x_ + S_t_, } U_{1} = β(_u_{1} - _v_)/(P_u_{1} + S), } (I) U_{2} = _u_{2}/(P_u_{1} + S), } U_{3} = _u_{3}/(P_u_{1} + S). } Here we remember that (_u_{1}, _u_{2}, _u_{3}) are the velocities of any particle according to A's "space and clock" system, and that (U_{1}, U_{2}, U_{3}) are the velocities of the same point according to B's "space and clock" system. We have obtained the most general relations consistent with the facts that (1) they both employ Euclidean systems, related as described above, and (2) they agree in their judgments on the uniformity of velocity. We now compare their judgments on the magnitudes of velocities. Let the magnitude of the velocity of P be V according to A's judgment, and V′ according to B's’ judgment. Then V^2 = _u_{1}^2 + _u_{2}^2 + _u_{3}^2, V′^2 = U_{1}^2 + U_{2}^2 + U_{3}^2. Also we can put _u_{1} = _l_V, _u_{2} = _m_V, _u_{3} = _n_V, where (_l_, _m_, _n_) have nothing to do with the magnitude V, but simply depend on the direction of motion. In fact (_l_, _m_, _n_) are the "direction cosines" of the velocity according to A's judgment. By substituting in the above equation for V^2 we see that _l_^2 + _m_^2 + _n_^2 = 1. Now, substituting for (_u_{1}, _u_{2}, _u_{3}) in the equations (I) above, and squaring and adding, and eliminating _m_^2 + _n_^2 by the relation just found, we at once find V′^2 = ((β^2 - 1)V^2_l_^2 - 2β^2V_vl_ + β^2_v_^2 + V^2)/(PV_l_ + S)^2. It is thus seen that in general the relation of V′ to V depends on the direction cosine _l_. Now _l_ is the cosine of the angle which the direction of the velocity V makes with O_x_, according to A's judgment. The meaning of this relation is, that if A discharges, from guns at the point P, shells with a given muzzle velocity V according to his judgment, B will consider that their muzzle velocities are different from each other, except in the case of pairs of guns equally inclined to the axis OO′. Instances of this type of diversity of judgment can be noted any day by any one who looks out of the window of a railway carriage, and forgets that he is travelling. Now, suppose the velocity V′ bears a relation to the velocity V, which is independent of _l_. Then _l_ must disappear from the above formula. There are two conditions to be satisfied One condition is V^2 = β^2_v_^2/(β^2 - 1), or in a more convenient form β^2 = 1/(1 - _v_^2/V^2). The meaning of this condition is, that there is one, and only one, muzzle velocity V (according to A's judgment), namely, the muzzle velocity given by the above formula, which can have the property that B will judge that all the guns are firing in their diverse directions with one common muzzle velocity.