Page:SahaElectrodynamics.djvu/17

 The law of parallelogram of velocities hold up to the first order of approximation. We can put

$$U^{2}=\left(\frac{\partial x}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial t}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2}$$,

and

$$\alpha=\tan^{-1}\frac{w}{w_{\xi}}$$

i.e., $$\alpha$$ is put equal to the angle between the velocities v, and w. Then we have—

$$U=\frac{\left[(v^{2}+w^{2}+2vw\ \cos\ \alpha)-\left(\frac{vw\ \sin\ \alpha}{c}\right)^{2}\right]^{\frac{1}{2}}}{1+\frac{vw\ \cos\ \alpha}{c^{2}}}$$

It should be noticed that v and w enter into the expression for velocity symmetrically. If w has the direction of the &xi;-axis of the moving system,

$$U=\frac{v+w}{1+\frac{vw}{c^{2}}}$$

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 $$v = c - \chi$$, and $$w = c - \lambda$$ where $$\chi$$ and $$\lambda$$ are each smaller than c,

$$U=c\frac{2c-\chi-\lambda}{2c-\chi-\lambda+\frac{\chi\lambda}{c^{2}}}<c$$.

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,

$$U=\frac{c+v}{1+\frac{cv}{c^{2}}}=c$$