A Treatise on the Theory of Screws

316] THE GEOMETRICAL THEORY. 341 Let r) be any screw on which an impulsive wrench is imparted, and let a be the corresponding instantaneous screw, about which the body would have begun to twist had it been free. Draw the cylindroid through a and X, and choose on this cylindroid the screw p, which is reciprocal to p. Then p is the instantaneous screw about which the body commences to twist, in consequence of the impulsive wrench on y. For (-7, p) is a cylindroid of impulsive screws, and (a, X) are the corre- sponding instantaneous screws. As p is reciprocal to p, it belongs to the system of the fifth order. The corresponding impulsive screw must lie on (?7, p). The actual instantaneous motion could therefore have been produced by impulsive wrenches on 17 and p. The latter would, however, be neutralized by the reactions of the constraints. We therefore find that 17 is the impulsive screw, corresponding to a as the instantaneous screw. 316. Principal Screws of Inertia of Constrained Body. There is no more important theorem in this part of the Theory of Screws than that which affirms that for a rigid body, with n degrees of freedom, there are n screws, such that if the body when quiescent receives an impulsive wrench about one of such screws, it will immediately commence to move by twisting about the same screw. We shall show how the principles, already explained, will enable us to construct these screws. We commence with the case in which the body has two degrees of freedom. We take three screws, tj, %, %, arbitrarily selected on the cylindroid, which expresses the freedom of the body. We can then de- termine, by the preceding investigation, the three instantaneous screws, a, ß, 7, on the same cylindroid, which correspond, respectively, to the impulsive screws. Of course, if p happened to coincide with a, or £ with ß, or £ with y, one of the principal screws of inertia would have been found. But, in general, such pairs will not coincide. We have to show how, from the knowledge of three such pairs, in general, the two principal screws of inertia can be found. We employ the circular representation of the points on the cylindroid, as explained in § 50. The impulsive screws are represented by one system of points, the corresponding instantaneous screws are represented by another system of points. It is an essential principle, that the two systems of points, so related, are homographic. The discovery of the principal screws of inertia is thus reduced to the well-known problem of the discovery of the double points of two homographic systems on a circle.