4. The derived transformation equations.

4.1Area (s)

ds' = ds, or, in general, dsm = dsn
image (6)

Con-space area:
image (6a)

4.2Volume (image)

image or, in general, image
image (7)

Con-space volume (V):
image (7a)

4.3 Speed (v)

v' = v or, in general, vm = vn
To precede possible misunderstanding , the meaning of the speed v or v' is the following:
  1. When a body is moving at a speed v in the unprimed frame, the observer from the primed frame can perform the measurement of this speed (not the resultant speed) by the instruments breathing with the spacetime :
    1. from the primed frame, after the primed frame has been accelerated or braked, to eliminate the relative motion between the frames,
    2. or after the observer has accelerated, or braked, to eliminate the relative motion between the frames.
    In both cases the measurement shows the same magnitude of the speed, now designated as v'.
  2. If a body was moving at a speed v' in the primed frame, and the observer from the unprimed frame performed the similar measurement of the speed v, he would find out, that v' = v as well.
The speed in a frame M, measured by the instruments not breathing with space and time and calibrated for the time and length units of the N frame, may be expressed :
image (8)

and the con-spacetime speed calculated from con-spacetime length and time is defined by equation (5c).