A Treatise on the Theory of Screws

360 THE THEORY OF SCREWS. [332, 332. Co-reciprocal Correspondents in two Three-systems. If U be an instantaneous three-system and V the corresponding im- pulsive three-system it is in general possible to select one set of three co-reciprocal screws in U whose correspondents in V are also co-reciprocal. As a preliminary to the formal demonstration we may note that the number of available constants is just so many as to suggest that some finite number of triads in Ü ought to fulfil the required condition. In the choice of a screw a in Ü we have, of course, two disposable quantities. In the choice of ß which while belonging to U is further reciprocal to a there is only one quantity disposable. The screw belonging to U, which is reciprocal both to a and ß, must be unique. It is in fact reciprocal to five independent screws, i.e. to three of the screws of the system reciprocal to U, and to a and ß in addition. We have thus, in general, neither more nor fewer than three disposable elements in the choice of a set of three co-reciprocal screws a, ß, 7 in U. This is just the number of disposables required for the adjustment of the three correspondents v, £ f in 7 to a co-reciprocal system. We might, therefore, expect to have the number of solutions to our problem finite. We are now to show that this number is unity. Taking the six principal screws of inertia of the rigid body as the screws of reference, we have as the co-ordinates of any screw in U X«! + pß1 + vp, b-O-2 + nßz + vy.2, Xa6 + pß,. 4- vy6, where X, /a, v are numerical parameters. The co-ordinates of the corresponding screw in V are Pi (A.ax + pß2 + i/y,), P2 (Xa2 + pß.t + Ph + pßts + vy^), where for symmetry plt ... are written instead of + a, - a, + b, - b, &c. Three screws in U are specified by the parameters x; p', v'- x", p.", v"- X"', v"\ If these screws are reciprocal, we have 0 = (X'a, + p'ßx + (x"ai + or 0 = X'X"pa + p'p'pp + v'v"py + + . . _ + (XV'+ XV) + (^V'+ and two similar equations.