Try to calculate in more detail. What is the differential – is the limit of finite differences. We write the finite difference (eg, x): x1 = (vt'1 + x'1) / (1-v / c ) (4A) This is correct. We write x2. From a mathematical point of view – have changed the codes and everything. But from a practical point of view to measure x2 (to keep the synchronization clock to all the coordinates correspond to each other) can only signal that goes from S 'in S. This signal goes from x2 ', is the center coordinates of S', then goes to the center coordinates of S.
Interested parties can try to measure yourself. All literature on this dimension, describes just such a way. However, measuring this way corresponds to the minus sign in the velocity (in absolute values) in the formula (1A). Read more here: Lynn Redgrave. This means that the finite difference we do not record. Velocity addition formula is easily obtained from a comparison of size, such as line length L, in different inertial coordinate systems. Take the three inertial coordinate system S, S ', S'': S – rest, S'-moving S with velocity v, and S''- moving in S 'with velocity u'. Then the lengths respectively are L, L ', L''. If the initial time points of coordinates coincide (top line coincides with the center coordinates), then the coordinate of the end line coincides with its length. Coordinates of different coordinate systems (and consequently the end of the line) we do not associate with each other.