A Treatise on the Theory of Screws

309] THE GEOMETRICAL THEORY. 333 will, by the property of a principal screw of inertia, produce an instantaneous twist velocity about the same screw. But the two twist velocities so generated can, of course, only compound into a single twist velocity on some other screw of the cylindroid. We have now to obtain the geometrical relations characteristic of the pairs of impulsive and instantaneous screws on such a cylindroid. In previous chapters we have discussed the relations between impulsive screws and instantaneous screws, when the movements of the body are confined, by geometrical constraint, to twists about the screws on a cylindroid. The problem now before us is a special case, for though the movements are no other than twists about the screws on a cylindroid, yet this restriction, in the present case, is not the result of constraint. It arises from the fact that two of the six principal screws of inertia of the rigid body lie on the cylindroid, while the impulsive wrench is, by hypothesis, limited to the same surface. To study the question we shall make use of the circular representation of the cylindroid, § 50. We have there shown how, when the several screws on the cylindroid are represented by points on the circumference of a circle, various dynamical problems can be solved with simplicity and convenience. For example, when the impulsive screw is represented on the circle by one point, and the instantaneous screw by another, we have seen how these points are connected by geometrical construction (§ 140). In the case of the unconstrained body, which is that now before us, it is known that, whenever the pitch of an instantaneous screw is zero, the corre- sponding impulsive screw must be at right angles thereto (§ 301). In the circular representation, the angle between any two screws is equal to the angle subtended in the representative circle by the chord whose extremities are the representatives of the two screws. Two screws, at right angles, are consequently represented by the extremities of a diameter of the representative circle. If, therefore, we take A, B, two points on the circle, to represent the two screws of zero pitch, then the two points, P and Q, diametrically opposite to them, are the points indicating the corresponding impulsive screws. It is plain from § 287 that AQ and BP must intersect in the homographic axis, and hence the homographic axis is parallel to J Q and BP, and as it must contain the pole of AB it follows that the homographic axis XY must be the diameter perpendicular to AB. The two principal screws of the cylindroid X and Y are, in this case, the principal screws of inertia. Each of them, when regarded as an impulsive screw, coincides with its corresponding instantaneous screw. The diameter XY bisects the angle between AP and BQ.