A Treatise on the Theory of Screws

178 the theory of screws. [176- We see from this that the sum of the reciprocals of the pitches of three co-reciprocal screws is constant. This theorem will be subsequently generalised. 177. Locus of the feet of perpendiculars on the generators*. If p be the pitch of the screw of the three-system which makes angles a, ß, y with the three principal screws, it is then easy to show that, the equation of the screw is (/ > — «) cos a + z cos ß — y cos 7 = 0, — z cos a + (p — b) cos ß + x cos y = 0, + y cos a — æ cos ß + lp - c) cos 7 = 0. If perpendiculars be let fall from the origin on the several screws of the system, then if x, y, z be the foot of one of the perpendiculars x cos a + y cos ß 4- z cos 7 = 0. Eliminating cos a, cos ß, cos 7 from this equation and the two last of those above, we have æ y -z p-b +y -x = o, or (P-^)(p-c)® + ®(«2 + 2/2 + ^) + ^(&-c) = O; from this and the two similar equations we have, by elimination of p2 and p and denoting 4- y* + z2 by r2, x, (b + c) x, bcx + (b — c)yz + xr2 y, (c + a) y, cay + (c — a) zx + yr2 z, (a + 6) z, abz + (a-b)xy + zr2 = 0; multiplying the first column by r2 and subtracting it from the last, we have a?, (b + c) x, y, (c + a)y, \ z, (a + b) z, which may be written (a - b)2 x2y2 + (b - c)2 y2z2 + (c bcx + (b — c) yz cay + (c — a) zx abz + (a - b) xy = 0, '-:t' ' 1 - - a)2 z2x2 = (a — b)(b — c)(c — a) xyz. This Article is due to Professor C. Joly, ‘ On the theory of linear vector functions,’ Transac- twns of the Royal Irish Academy, Vol. xxx. pp. 601 and 617 (1895), where a profound discussion of Steiner s surface is given. See also by the same author ‘Bishop Law’s Mathematical Prize Examination,’ Dublin University Examination Papers, 1898.