Summing Velocities

Since there is a non-linear relation between the velocities defined in (3.1) and (3.2) and the true velocity obtained from the full relativistic expressions, one cannot simply add the numbers when summing various velocities. If we assume that $V_{0}$ is the sum of $V_{1}$ and $V_{2}$, summation should be performed according to the following rules:

$$V_{0}^{r} = V_{1}^{r} + V_{2}^{r} - \frac{V_{1}^{r}\; V_{2}^{r}}{c}$$ (3.3)

$$V_{0}^{o} = V_{1}^{o} + V_{2}^{o} - \frac{V_{1}^{o}\; V_{2}^{o}}{c}$$ (3.4)

The formula for the redshift becomes:

$$(1 + z_{0}) = (1 + z_{1})(1 + z_{2})$$ (3.5)