A Treatise on the Theory of Screws

49] SCREW CO-ORDINATES. 43 Hence by subtracting the several formulae (i) from the formulae (ii) we obtain !)„ («5 + aB) - z„ (a3 + a4) = a (a, - a.) - a (0, - Ö.) 0» ( + ßs}2], or p2pa + 2p-naß + pß = p [p2 + 2p cos (aß) + 1}, or p2 (P* -p) + %P - P c°s (a/3)} + pß - p = 0. For the principal screws p is to be determined so that p shall be a maxi- mum or a minimum (§ 18), whence the equation for p is {^■«ß -p cos (aß)}2=(pa -p) (pß-p), or p2 sin2 (aß) + p (2oto/3 cos (aß) -pa- pß) + papp - = 0. The roots of this quadratic are the required values of p. The quadratic may also receive the form o = (p - pa) (p - pp) sin2 (aß) + I daß sin (aß) cos (aß) (pa + pp - 2p) - 1 (?« -Pß)2 cos3 (aß) ~ 1 sin3 (aß)> where daP is the shortest distance of a and ß.