Page:The principle of relativity (1920).djvu/156

 be called the Velocity-vector, and the Acceleration-vector of the substantial point at P. Now we have

c^2 [.t]^ - [.x]^2 - [.y]^2 - [.z]^2 = c^2                         } c^2 [.t][..t] - [.x][..x] - [.y][..y] - [.z][..z] = 0  },

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 c^2/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 = [infinity].

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