A Treatise on the Theory of Screws

_______ ________ _____ ________ SCREW CO-ORDINATES. ________________________________________________________ 44] ____ ____ 39 we have similarly we thus find 2<ta1 = ( a+/>«)X" ——«'"y), - 2aa2 - (- a + pa) X" - (p'z' — v"y'\ X" = aj + a.,; v"y'- pz' = a(a\ — a2)-pa(a, 4- a2). In like manner we obtain two similar pairs of equations for the required equations of the screw a, (a-, + ?«) y - (a3 + a,) z = a (sq - a,) - p« (a, + a2), ' (a, + a3) z - (a6 + a„) x = b («3 - a4) - pa (a3 + at), - (a, + a4) x — (a, + a2) y = c (a5 - ab) — j»o (a5 + a6). ....... (i)- The expressions on the right-hand side of these equations are the co- ordinates of the extremity of a vector from the origin of length equal to the perpendicular distance of a from the centre, and normal to the plane containing both a and the origin. The co-ordinates of the foot of the perpendicular from the origin on the screw a are easily shown to be x - (as - a„) (a3 + a4) c - (a5 + a«) (a» - a4) b, y={ocl- a3) («6 + a«) a - (a, + a3) (a5 - ct6) c, z = (a3 - a,) (a, 4- a,) b - (a3 + a4) (a, - a3) a. 44. A Screw of Infinite Pitch. The conception of the screw co-ord.ina.tcs hs defined in § 4*1 require special consideration in the case oi a screw of infinite pitch. Consider a wrench on such a screw. If the intensity of the wrench be one unit, thou the moment of the couple which forms part of the wrench is infinite. Ah the pitches of the screws of reference or any of those pitches are not in general to be infinite, it follows that the wrench of unit intensity on a screw of infinite pitch must have for its components on one or more of the screws of reference wrenches of infinite intensity. If therefore an a2, ... a6 be the co-ordinates of a wrench of infinite pitch, it is essential that one or more of the quantities a1; a.,, ... a6 shall be infinite. In the case where the screws of reference form a canonical system we can obtain the co-ordinates as follows: < (V. + a) cos (al) - (41 sin (al). _ _ (pa - a) cos (al) - sin (al) «> = ------------2cT ’ 2 " ’ - 2a