A Treatise on the Theory of Screws

249] HOMOGRAPHIC SCREW SYSTEMS. 265 seven pairs thus give 35 relations which suffice to determine linearly the ratios of the coefficients. The screw ß corresponding to any other screw a is completely determined ; we have therefore proved that When seven corresponding pairs of screws are given, the two homographic screw systems are completely determined. A perfectly general way of conceiving two homographic screw systems may be thus stated:—Decompose a wrench of given intensity on a screw a into wrenches on six arbitrary screws. Multiply the intensity of each of the six component wrenches by an arbitrary constant; construct the wrench on the screw ß which is the resultant of the six components thus modified; then as a moves into every position in space, and has every fluctuation in pitch, so will ß trace out the homographic screw system. It is easily seen that in this statement we might have spoken of twist velocities instead of wrenches. 249. Homographic »-systems. The seven pairs of screws of which the two systems are defined cannot be always chosen arbitrarily. If, for example, three of the screws were co- cylindroidal, then the three corresponding screws must be co-cylindroidal, and can only be chosen arbitrarily subject to this imperative restriction. More generally we shall now prove that if any n +1 screws belong to an «-system (§69), then the n +1 corresponding screws will also belong to an «-system. If n + 1 screws belong to an n-system it will always be possible to determine the intensities of certain wrenches on the n + 1 screws which when compounded together will equilibrate. The conditions that this shall be possible ai-e easily expressed. Take, for example, n = 3, and suppose that the four screws a, ß, y, 8 are such that suitable wrenches on them, or twist velocities about them, neutralize. It is then obvious (§ 76) that each of the determinants must vanish which is formed by taking four columns from the expression otj, a3, a3) a4, a5, a6 ßl, ßi, ßs, Ä, ß5, ß6 71, 72, 7s> 7<> 7s> Vs 8« It is, however, easy to see that these determinants will equally vanish for the corresponding screws in the homographic system ; for if we take as screws of reference the six common screws of the two systems, then we have at once for the co-ordinates of the screw corresponding to a (11) au (22) a2, (33) a8, (44) a4, (55) a6, (66) a6.