Page:The principle of relativity (1920).djvu/147



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 c^{2}t^{2} - x^{2} = 1, 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´ = (1/c) ; then the hyperbola is again expressed in the form c^{2}t´^{2} - x´^{2} = 1, t´ > 0 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 (1/c) 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=[infinity]. In this state of affairs, and since G_{c} is mathematically more intelligible than G_{[infinity]}, 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_{[infinity]}, but in fact also for a group G_{c}, where c is finite, but yet