A Treatise on the Theory of Screws

184] FREEDOM OF THE THIRD ORDER. 185 a1; ...a6; ßx, ... ße', 71,... y6. Then if X, /a, v be three variable parameters, the co-ordinates of the other screws of the three-system will be Xai + ^i + rVi, Xa.> + /z/3., + vy2, ••• Xa8 + pß6 + vy6. We shall denote the pitch of this screw by p, and from § 43 we have for the equations of this screw with reference to the associated Cartesian axes: o = + (X«5 + pßs + vy5 + Xa„ + pß6 + vye) y — (Xa3 + pß3 + vy3 + Xa4 + pßt + vyA) z — (Xaj + /x/3j + vy, — Xa2 — pß., — i*y2) a — (X«i + pßi + vy-i + ÅÆ + pß3 + vyp p, with two similar equations. From these we eliminate X, p, v and the determinant thus arising admits of an important reduction. To effect this we multiply it by the determinant «1 + a2, «3 + a4, a6 + a6 I. ßl + ß-2> ßs + ßi> ßs + ßs | 7i + y2 > ys + 74, 75 + 70 For brevity we introduce the following notation: P - ® [(Ä + A) (?3 + 74) - (Ä + ßß (?6 + 7e)] + y [(Ä + A) (75 + 7s) - {ß-> + &) (yj + v»)] + 2 [(A + Ä) (?i + 7a) — {ßl + /3a) (73 + 7a)] > with similar values for Q and R by cyclical interchange. We also make = a (a, + a2) {ß2 -ß.2) + b (a3 + a4) {ß3 - ßt) + c (a, + a6) (ßs - ß6), Lßa = a{ßi + ß.2) (ax - a.,) + b(ß3 + ßß (a3 - a4) + c {ß5 + ß,) (a6 - a6), with similar values for Lay, Lya, Lßy, Lyß by cyclical interchange. The equation to the family of pitch quadrics is then easily seen to be 0 = 1 Pa~p , -P +Laß-pcos{aß), + Q + Lay—peos{ay) . + R + Lßa-p cos (aß), Pß-P , -P + Lßy-pcoa(ßy) \ - Q +Lya-pcos(ay), + P 4- Lyß - p COS {ßy), Py~P If the three given screws a, ß, y had been co-reciprocal, then as Laß + Lßa = 2m aß = 0, it follows that Laß and Lßa only differ in sign, so that if P' = P + Lyß-, Q’^Q + Lay- R’ = R + Lßa,