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The conformal transformations of a space of four dimensions.

terms of the coordinates (x, y, z), and the constants of the standard wave front. Then V satisfies the differential equation[1]

and is, in fact, the characteristic function introduced by Hamilton. If it is expressed as a function of x, y, z, and the coordinates of the initial point , it is the Eikonal according to the nomenclature of Bruns.[2]

Since V is proportional to the time this differential equation may be replaced by

where C is the velocity of radiation at the point (x, y, z).

Now suppose that the surfaces t = const, are obtained by solving an equation

for t; then, since


the function F must satisfy the differential equation

Confining ourselves to the case in which C is constant, we may use the results of § 2 to obtain new solutions of this differential equation.


be the formulæ giving a transformation which enables us to pass from one solution of the above equation to another; then

when expressed in terms of x, y, z, t, is a second solution of the equation, and if the equation

be solved for t, the surfaces t = const, will form a system of parallel wave

  1. See Herman's Optics, p. 253.
  2. Cf. Schwarzschild's Untersuchungen zur Geometrischen Optik, Göttingen Abhandlungen (2), 4.