A Treatise on the Theory of Screws

KMNMMMMMMiM NMHM 272 THE THEORY OF SCREWS. [258, 258. Corresponding Screws defined by Equations. It is easy to state the matter analytically, and for convenience we shall take a three-system, though it will be obvious that the process is quite general. Of the six screws of reference, let three screws be chosen on the three- system, then the co-ordinates of any screw on that system will be alt a2, «3, the other three co-ordinates being equal to zero. The co-ordinates of the corresponding screw ß must be indeterminate, for any screw of a four-system will correspond to ß. This provision is secured by ßit ßs, ßs remaining quite arbitrary, while we have for ßlt ß2, ß3 the definite values, Ä = (11) ax + (12) a2 + (13) as, ß2 — (21) oti + (22) a2 + (23) a3, ßs = (31) a, + (32) a2 + (33) as. If we take ßit ß5, ßa all zero, then the values of ßT, ß2, ßs, just written, give the co-ordinates of the special screw belonging to the three-system, which is among those which correspond to a. As a moves over the three-system, so will the other screw of that system which corresponds thereto. There will, however, be three cases in which the two screws coincide; these are found at once by making ßl ~ Pai > ß‘2 = Paz j ßs= Pas> whence we obtain a cubic for p. It is thus seen that generally n screws can be found on an n-system, so that each screw shall coincide with its correspondent. As a dynamical illustration we may give the important theorem, that when a rigid body has n degrees of freedom, then n screws can always be found, about any one of which the body will commence to twist when it receives an impulsive wrench on the same screw. These screws are of course the principal screws of inertia (§ 84). 259. Generalization of Anharmonic Ratio. We have already seen the anharmonic equality between four screws on a cylindroid, and the four corresponding screws; we have also shown a quasi anharmonic equality between any eight screws in space and their cor- respondents. More generally, any n + 2 screws of an «-system are connected with their w + 2 correspondents, by relations which are analogous to an- harmonic properties. The invariants are not generally so simple as in the eight-screw case, but we may state them, at all events, for the case of n = 3. Five screws belonging to a three-system, and their five correspondents