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PRINCIPLE OF RELATIVITY

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

*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 in the direction of* MP.

If are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.

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