A Treatise on the Theory of Screws

337] and similarly, VARIOUS EXERCISES. 365 —cos (y£) =------------cos (ßt;), cos(8£) cos(yf) Wb/’ -,1 cos (a£) =------cos (yj?), cos(yø v cos (a?;) v/ ’ whence we obtain cos (af) cos (/S^) cos (7^) + cos (af) cos (/3f) cos (y?;) = 0, for it is shown in § 283 that p^ cos (ap) or the other similar expressions can never be zero. 336. Instantaneous Screw of Zero Pitch. Let a be an instantaneous screw of zero pitch. Let two of the canonical co-reciprocals lie on a, then the co-ordinates of a are I, 1 0, 0, 0, 0. The co-ordinates of the impulsive screw y are given by the formulae of § 326 which show that n 2^0 2w„ Vi + % = 0; % + y4 =--------; + % = + . 6 6 We thus have («1 + a2) (ph + %) + (a3 + «4) (% + Tji) + (a5 + a6) (% + = 0, which proves what we already knew, namely, that a and y are at right angles (§ 293). We also have y» (% + %) + z„ (i?5 + - 0, which proves the following theorem : If the instantaneous screw have zero-pitch then the centre of gravity of the body lies in the plane through the instantaneous screw and perpendicular to the impulsive screw. 337. Calculation of a Pitch Quadric. If a, ß, y be three instantaneous screws it is required to find with respect to the principal axes through the centre of gravity, the equation to the pitch quadric of the three-system which contains the three impulsive screws corre- sponding respectively to a, ß, y. The co-ordinates of these screws are expressed with reference to the six principal screws of inertia. We make the following abbreviations: A = a3 (a,2 - a22) + b3 (a32 - a,2) + c3 (a52 - a62), B = a3 (ß^-ß.ß) + b3 (ß^ - ß*) + c3 (ß* - ßß C=a3 (7? - 7/) + b3 (y33 - 7?) + cs (7.,2 - 7(j2);