_i.e._, the ‘_Velocity-vector_’ is the time-like vector of unit measure in the direction of the world-line at P, the ‘_Acceleration-vector_’ at P is normal to the velocity-vector at P, and is in any case, a space-like vector. Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a ‘fore-cone’ and an ‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at P (_vide_ fig. 3). If M be the centre of this hyperbola, then we have to deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is _a vector of magnitude_ _c²_/ρ _in the direction of_ MP. If [.._x_], [.._y_], [.._z_], [.._t_] are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞. IV In order to demonstrate that the assumption of the group G_{_c_} for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of “Thermodynamics and Radiation,”[30] for “Electromagnetic phenomena”,[31] and finally for “Mechanics with the maintenance of the idea of mass.” For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (_x_, _y_, _z_, _t_), where the velocity-vector is ([._x_], [._y_], [._z_], [._t_]), then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group G_{_c_} is undoubtedly permissible. These theorems lead us to the following simple rule; _if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components_ [._t_]X, [._t_]Y, [._t_]Z, [._t_]T, _where_ T = 1/_c²_ ([._x_]/[._t_] X + [._y_]/[._t_] Y + [._z_]/[._t_] Z) = 1/_c²_ (_the rate of which work is done at the world-point_), _remains unaltered_. This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a _moving force-vector at_ P.