APPENDIX

73

transformation in which (*y, z*) remain unaltered. Let us draw the cross section of the upper sheets with the plane of the *x*-and *t*-axes, *i.e.*, the upper half of the hyperbola , with its asymptotes (*vide* fig. 1).

Then let us draw the radius rector OA', the tangent A' B' at A', and let us complete the parallelogram OA' B' C'; also produce B' C' to meet the *x*-axis at D'. Let us now take Ox', OA' as new axes with the unit measuring rods OC' = 1, OA'= ; then the hyperbola is again expressed in the form and the transition from (*x, y, z, t*) to (*x' y' z' t*) is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points; then we get a group of transformation depending only on *c*, which we may denote by G_{c}.

Now let us increase *c* to infinity. Thus becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the *x*-axis, the asymptotic angle becomes a straight one, and every special transformation in the limit changes in such a manner that the *t*-axis can have any possible direction upwards, and *x' *more and more approximates to x. Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group G_{c}, with the value of *c* = ∞. In this state of affairs, and since G_{c} is mathematically more intelligible than G_{∞}, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G_{∞}, but in fact also for a group G_{c}, where *c* is finite, but yet