Annus MirabilisEinstein in 1905
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special relativity

The equation explained

cos⁡φ′=cos⁡φ−vV1−vVcos⁡φ.\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi} .
In wordsThe cosine of the ray's angle in the moving system is the cosine of its angle in the stationary system less the speed over the velocity of light, divided by one minus that ratio times the same cosine.OverviewThe law of aberration in its most general form, as the paper calls it: how the direction of a ray, measured from the direction of motion, differs between the stationary and the moving observer.Full explanationThe paper takes the equation for a′a' and writes both cosines as angles: φ\varphi is the angle between the ray and the line of motion in the source's system, φ′\varphi' the angle between the wave normal (the direction of the ray) in the moving system and the same line. Aberration is the change from φ\varphi to φ′\varphi'. It vanishes for rays along the motion and is largest for rays near right angles to it. The light the moving observer receives comes from a direction shifted towards the direction of motion, which is what an astronomer observes as the aberration of starlight.Every stepa′=a−vV1−avVa' = \frac{a - \frac{v}{V}}{1 - a \frac{v}{V}}The first new direction cosine, from p. 911.a=cos⁡φ,a′=cos⁡φ′a = \cos \varphi, \quad a' = \cos \varphi'The direction cosine along the motion is the cosine of the angle between the ray and the line of motion, in each system.cos⁡φ′=cos⁡φ−vV1−vVcos⁡φ\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi}Substitute. For φ=0\varphi = 0 this gives cos⁡φ′=1\cos \varphi' = 1: a ray along the motion keeps its direction.Also

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