Page:BatemanConformal.djvu/16

From Wikisource
Jump to navigation Jump to search
This page has been validated.
1908.]
85
The conformal transformations of a space of four dimensions.

Consider a ray of light travelling from in a direction (l, m, n) with velocity c. At time t the wave has reached a point (x, y, z) on the ray, where

The corresponding point (X, Y, Z) derived from this by the transformation also travels along a straight line, for

,

where

The corresponding ray thus passes through the inverse point on the inverse surface at which it may be supposed to be incident. Its direction cosines (L, M, N) are connected with those of the former ray by means of the equations

These relations establish a correspondence between the sheafs of rays through the points , respectively. This correspondence is such that the angle between two rays (l, m, n), (l', m', n') is equal to the angle between the two corresponding rays (L, M, N), (L', M', N'), for we have identically

Since the transformation enables us to derive the surfaces which are parallel to one surface from the surfaces which are parallel to the inverse surface, it is natural to expect that the above relation between the direction cosines will make the normals to the two surfaces correspond.