Annus MirabilisEinstein in 1905
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special relativity

The equation explained

U=(v2+w2+2vwcos⁡α)−(vwsin⁡αV)21+vwcos⁡αV2.U = \frac{\sqrt{(v^2 + w^2 + 2 v w \cos \alpha) - \left( \frac{v w \sin \alpha}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}} .
In wordsThe resulting speed is the square root of the square of the speed of the moving system plus the square of the speed of the point in it plus twice their product times the cosine of the angle between them, less the square of their product times the sine of that angle over the speed of light; all divided by one plus their product times that cosine over the square of the speed of light.OverviewThe general rule for compounding two velocities that make an angle with each other. Under the root sits what the old parallelogram rule would give, and beside it a subtraction and a denominator that are new; both vanish when the speeds are small against light.Full explanationThe two components of ww are wcos⁡αw \cos \alpha and wsin⁡αw \sin \alpha, and putting them into the two rates of the display before and adding the squares gives this after a simple calculation. Under the root, the first three terms are exactly the parallelogram law, the square of the sum of two vectors making the angle α\alpha. What is new is the term subtracted from them and the denominator, and each carries the speed of light in a square. The paper then remarks on the form: it is noteworthy that vv and ww enter the expression for the resulting velocity in a symmetric way, so it makes no difference which of the two is called the motion of the system and which the motion of the point in it. Nothing here restricts α\alpha, so this one formula covers the two special cases that follow, the same direction and the case of light.Every stepdxdt=wcos⁡α+v1+vwcos⁡αV2,dydt=1−(vV)21+vwcos⁡αV2wsin⁡α\frac{d x}{d t} = \frac{w \cos \alpha + v}{1 + \frac{v w \cos \alpha}{V^2}}, \quad \frac{d y}{d t} = \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}} w \sin \alphaThe two rates with the components of ww written by the angle.U2=(wcos⁡α+v)2+(1−v2V2)w2sin⁡2α(1+vwcos⁡αV2)2U^2 = \frac{(w \cos \alpha + v)^2 + \left( 1 - \frac{v^2}{V^2} \right) w^2 \sin^2 \alpha}{\left( 1 + \frac{v w \cos \alpha}{V^2} \right)^2}Their squares added. The denominator is the same for both and comes out squared.(wcos⁡α+v)2+w2sin⁡2α=v2+w2+2vwcos⁡α(w \cos \alpha + v)^2 + w^2 \sin^2 \alpha = v^2 + w^2 + 2 v w \cos \alphaThe numerator without the correcting term: the squares of the cosine and sine parts add to w2w^2, and the cross term is twice the product with the cosine. This is the parallelogram law.−v2V2w2sin⁡2α=−(vwsin⁡αV)2- \frac{v^2}{V^2} w^2 \sin^2 \alpha = - \left( \frac{v w \sin \alpha}{V} \right)^2The remaining piece of the numerator, which is what the square root in the sideways rate leaves behind.U=(v2+w2+2vwcos⁡α)−(vwsin⁡αV)21+vwcos⁡αV2U = \frac{\sqrt{(v^2 + w^2 + 2 v w \cos \alpha) - \left( \frac{v w \sin \alpha}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}}The square root of the whole. The printed form, symmetric in the two speeds.Also

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