is of order ε², but there is one type of transformation in which it is of order ε³ at least. The formulae of transformation are then
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![{\displaystyle x'=x+\epsilon \left[2x(lx+my+nz-t)-l\left(x^{2}+y^{2}+z^{2}-t^{2}\right)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/383695fad3e47657ff3c818e84aad189ed3cf0c4)
![{\displaystyle y'=y+\epsilon \left[2y(lx+my+nz-t)-m\left(x^{2}+y^{2}+z^{2}-t^{2}\right)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee75e4bb5903d872d368e601251eeec9af69df4a)
![{\displaystyle z'=z+\epsilon \left[2z(lx+my+nz-t)-n\left(x^{2}+y^{2}+z^{2}-t^{2}\right)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e643f50f5c3839f765e3c6c5cc85e1324f1bfa76)
![{\displaystyle t'=t+\epsilon \left[2t(lx+my+nz-t)-\left(x^{2}+y^{2}+z^{2}-t^{2}\right)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43eb5a3e61501aa94b41495ac995ddae25f35981)
where
and signify that (x', y', z'), (x, y, z) are successive positions of a particle which is moving with the velocity of light.
Real spherical wave transformations may be obtained geometrically in the following way. Let the space time point (x, y, z, t) be represented by a sphere of radius t having its centre at the point (x, y, z). Then, if we apply a real conformal transformation of space to these representative spheres, the new set of spheres may be taken as the representative spheres of a new set of space time points (x', y', z', t'). These are connected with the original set by a system of equations which define a spherical wave transformation. This theorem has already been established in the case of an inversion,[1] and since any real conformal transformation of space can be built up from inversions, it follows that the general transformation obtained in the above way is a spherical wave transformation.
It should be noticed that the general spherical wave transformation cannot be obtained in this way, because t' = 0 always corresponds to t = 0. If, however, we combine these transformations with the real spherical wave transformation obtained by increasing or decreasing the radii of all the representative spheres by the same amount, it is possible to obtain any spherical wave transformation by a suitable combination. The proof of this will be left to the reader.
When we use the representative spheres the differential equation
admits of a very simple interpretation, as it implies that the two consecutive representative spheres specified by (x, y, z, t) and (x+dx, y+dy, z+dz, t+dt) touch one another internally.
