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» (<A 4- a2) - x„ (a5 + a6) = b (a3 - aj - b (6, - 04) - x„ («3 + a4) - y0 (a, + a2) = c (a5 - a„) - c (05 - 06) (iiii). The six equations (iii) and (iiii) determine 0,,... 06 in terms of a„... a6. 49. Principal Screws on a Cylindroid. If two screws are given we determine as follows the pitches of the two principal screws on the cylindroid which the two given screws define. Lot a and ß be the two given screws. Then the co-ordinates of these screws referred to six canonical co-reciprocals are «i,... a6 and ß„... ßs. The co-ordinates of any other screw on the same cylindroid are propor- tional to potj + ß,, pa2 ß2,... pot6 + ß« j when p is a variable parameter. The pitch p of the screw so indicated is given by the equation 41) a (pa, + ß,y - a (pa2 + ß2)2 + b (pa,, + ß3)2 - b (pa, + ß,)2 + c (pas + ßß2 - c (pa, + ß,)2 a p [{p (a, + a.,) + ß, + ß-?\2 + p {(ot3 + «,) + ß3 + + p + æ«) + ßr> + 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.