Page 188 of 212

Mathematically, we can characterise the generalised Lorentz

transformation thus:

It expresses x′, y′, x′, t′, in terms of linear homogeneous

functions of x, y, x, t, of such a kind that the relation

x′2 + y′2 + z′2 - c2t′2 = x2 + y2 + z2 - c2t2 (11a).

is satisficd identically. That is to say: If we substitute their

expressions in x, y, x, t, in place of x′, y′, x′, t′, on the

left-hand side, then the left-hand side of (11a) agrees with the

right-hand side.

APPENDIX II

MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")

(SUPPLEMENTARY TO SECTION XVII)

We can characterise the Lorentz transformation still more simply

if we introduce the imaginary

image031

in place of t, as time-variable. If, in accordance with this, we

insert

image050

and similarly for the accented system K′, then the condition

which is identically satisfied by the transformation can be

expressed thus:

x1′2 + x2′2 + x3′2 + x4′2 = x12 + x22 + x32 + x42 (12).