special relativity

The equation explained

\alpha = \operatorname{arctg} \frac{w_y}{w_x} ;

In wordsThe angle is the one whose tangent is the point's speed across the motion over its speed along the motion.OverviewThe direction the point travels in the moving system, given as the angle between its velocity and the velocity of that system. It is the third and last quantity the composition formula needs.Full explanationThe angle is defined by the ratio of the two components of the point's velocity in the moving system, and the sentence at the top of the next page says what it is for:

\alpha

is then to be regarded as the angle between the velocities

v

and

w

. With

w

and

\alpha

in hand the two components can be written

w \cos \alpha

along the motion and

w \sin \alpha

across it, which is what turns the two equations of the display before into one formula for the resulting speed. The formula that follows is symmetric in

v

and

w

, and this angle is why: it is the angle between them, and it does not privilege either.Every step

\alpha = \operatorname{arctg} \frac{w_y}{w_x}

As printed, with the components written in the letters of the axes. The tangent of the angle is the ratio of the component across the motion to the component along it.

w_\xi = w \cos \alpha, \quad w_\eta = w \sin \alpha

The same statement read backwards, which is the form the next display uses: the two components of a speed

w

making the angle

\alpha

with the direction of the motion.AlsoOverviewFull explanationEvery step

Read it in the paper