The law of parallelogram of velocities hold up to the first order of approximation. We can put
U^2 = ([part]x/[part]t)^2 + ([part]y/[part]t)^2, w^2 = w_{ξ}^2 + w_{η}^2,
and α = tan^{-1} w/w_{ξ}
i.e., α is put equal to the angle between the velocities v, and w. Then we have—
U = [(v^2 + w^2 + 2vw cos α) - (vw sin α/c)^2]^{1/2}/(1 + vw cos α/c^2)
It should be noticed that v and w enter into the expression for velocity symmetrically. If w has the direction of the ξ-axis of the moving system,
U = (v + w)/(1 + vw/c^2)
From this equation, we see that by combining two velocities, each of which is smaller than c, we obtain a velocity which is always smaller than c. If we put v = c - χ, and w = c - λ, where χ and λ are each smaller than c,
U = c(2c - χ - λ)/(2c - χ - λ + χλ/c^2) < c.[1]
It is also clear that the velocity of light c cannot be altered by adding to it a velocity smaller than c. For this case,
U = (c + v)/(1 + cv/c^2) = c.
- ↑ Vide Note 12.
