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General Formula for Sum of Velocities

Does anyone have a nice general formula for the sum of velocities?

We know that the sum of two velocities $$x,y$$ is $$\dfrac{x+y}{1+\dfrac{xy}{c^2}}$$ where $$c$$ is the speed of lights. I went ahead and calculated the sum of three velocities $$x,y,z$$, expecting to get something like $$\dfrac{x+y+z}{1+\dfrac{xyz}{c^3}}$$ but instead I got $\dfrac{x+y+z+\dfrac{xyz}{c^2}}{1+\dfrac{xy+yz+zx}{c^2}}$. I went further to calculate the sum of $$4$$ velocities $$w,x,y,z$$, to get the ugly expression $\dfrac{w+x+y+z+\dfrac{wxy+xyz+yzw+zwx}{c^2}}{1+\dfrac{wx+wy+wz+xy+xz+yz}{c^2}+\dfrac{wxyz}{c^4}}$

So can anyone else generalize? I do not see an obvious pattern yet, except for the fact that those different terms look suspiciously like something in Vieta's Formulas...

Note by Daniel Liu
3 years, 1 month ago