The proper motion of a star is determined through Doppler shift of lines in the spectrum of the star, and has to be determined for each star individually. You cannot just do some pythagorean math to get the proper motion of another star. The proper motion is the "true" motion of the star in space, where one takes out the effects of the orbit of the Earth and the rotation of the Earth etc etc.

Also, I haven't got a clue how you come up with that equation above, the square root of "Right Ascention" squared plus "Declination" squared? At least to have a result in arcseconds per year you should have a Δ or something in there.

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