A Treatise on the Theory of Screws

284 THE THEORY OF SCREWS. [268- where in which -^123 — 2 Pi’ sin2 (1, 2) Q___ \f -^123 PlPiPi 1, cos (12), cos (13) cos (12), 1, cos (23) cos (13), cos (23), 1 If by 8 (123) we denote the scalar of the product of three unit vectors along 1, 2, 3, then it is easy to show that ^(123) = ^. We thus obtain the following three relations between the pitches and the angular directions of the six screws of a co-reciprocal system*, Pt Pl s^(l, 2, 3) 0 PiP^P-i The first of these formulae gives the remarkable result that, the sum of the reciprocals of the pitches of the six screws of a co-reciprocal system is equal to zero. The following elegant proof of the first formula was communicated to me by my friend Professor Everett. Divide the six co-reciprocals into any two groups A and B of three each, then it appears from § 174 that the pitch quadric of each of these groups is identical. The three screws of A are parallel to a triad of conjugate diameters of the pitch quadric, and the sum of the reciprocals of the pitches is proportional to the sum of the squares of the conjugate diameters (§ 176). The three screws of B are parallel to another triad of conjugate diameters of the pitch quadric, and the sum of the reciprocals of the pitches, with their signs changed, is proportional to the sum of the squares of the conjugate diameters. Remembering that the sum of the squares of the two sets of conjugate diameters is equal, the required theorem is at once evident. * Proceedings of the Royal Irish Academy, Series hi. Vol. i. p. 375 (1890). A set of six screws are in general determined by 30 parameters. If those screws be reciprocal 15 conditions must be fulfilled. The above are three of the conditions, see also § 271.