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1908.]
75
The conformal transformations of a space of four dimensions.

Accordingly, from a solution V = F(x, y, z, w) of the equation

we may derive a second solution

and from the solution U = f(x, y, z, w) of

we may derive another solution

Putting w = ict, where c is the velocity of light and t is the time, the equations take the well known form[1]

and the transformation may be written

where now

The study of this transformation will be taken up later.


  1. It may be mentioned here that Lorentz's fundamental equations of the electron theory, viz.,

    may be reduced to a symmetrical form by writing s = ict and putting

    The four mutually orthogonal vectors (A, B, C, D) whose components are respectively

    (0, r, -q, -p), (-r, 0, p, -q), (q, -p, 0, -r), (p, q, r, 0)

    satisfy the equations

    where

    if

    Again, if we put

    and introduce four new vectors whose components are respectively

    (S, -Z, Y, -X) (Z, S, -X, -Y) (-Y, X, S, -Z) (X, Y, Z, S),

    we find

    where

    Finally, if X, Y, Z, S can be derived from a potential function n so that

    we can form four mutually orthogonal vectors θ, Φ, ψ, χ whose components are respectively

    (n, r, -q, -p). (-r, n, p, -q), (q, -p, n, -r), (p, q, r, n),

    and the equations then take the simple form