A Treatise on the Theory of Screws

228 THE THEORY OF SCREWS. [217- Of these the four screws with pitches - a, - b, + c, - c respectively are each reciprocal to the cylindroid. Each of these four screws must thus belong to the four-system. Further these four screws are co-reciprocal. It (?!...#„ be the six co-ordinates of a screw in the four-system referred to these canonical co-reciprocals, then we have e, = o, e3 = 0. For , but as the first screw of reference belongs to the reciprocal cylindroid we must have = 0. In like manner w39 = 0, and therefore = O and 03=O are the two linear equations which specify this particular four-system. The pitch of any screw on the four-system expressed in terms of its co-ordinates is -ae^-bd^ +_C{e^ - ^2 + ^ + (05 + ^ ’ of which the four stationary values are — a, — b, + oo , — oo . We may remark that if the four co-ordinates here employed be taken as a system of quadriplanar co-ordinates of a point we have a representation of the four-system by the points in space. Each point corresponds to one screw of the system. The screws of given pitch pe are found on the quadric surfaces Ü +peV = 0, where U = 0 is the quadric whose points correspond to the screws of zero pitch and where V= 0 is an imaginary cone whose points correspond to the screws of infinite pitch. Conjugate points with respect to U = 0 will cor- respond to reciprocal screws. A plane will correspond to a three-system and a straight line to a two-system. The general theorem proved in § 214 states that when 3 is a screw of stationary pitch in an n-system to which any other screw </> belongs, then = Pe cos 0</>. Let us now take a four-system referred to any four co-reciprocals and choose for <f> in the above formula each one of the four co-reciprocals in succession, we then have pA =pg {0, + 02 cos (12) + 0.3 cos (13) + 3, cos (14)}\ pA =pe {#i cos (12) 4- 02 + 03 cos (23) + 04 cos (24)} ^ö3=^[^cos(13) + ö2cos(23) + 03 + 04 cos (34)} — Pe {^i cos (14) + 02 cos (24) + 03 cos (34) + 04] , Eliminating 04, 02, 03l 04 we deduce a biquadratic for pg. But we have