A Treatise on the Theory of Screws

320 THE THEORY OF SCREWS. [299- necessity for this arrangement is thus shown. If not constant, then there would generally have been some screw £ for which was zero. In this case, of course, would be generally zero also. But 7 and y being both given, this is of course not generally true. The only escape is for -srof to be constant. 300. A difficulty removed. Given a and ß and £ and also 7, then the plane of £ is determined from the equations of the last article. As and are constant, both a and ß must be parallel to the plane already considered. But as an impulsive screw could not be reciprocal to an instantaneous screw, it would seem that must never be zero, but this condition can only be fulfilled by requiring that f must be parallel to the same plane. Whence a, ß, 7 must be parallel to the same plane. But these three screws are quite arbitrary. Here then would seem to be a contradiction. The difficulty can be explained as follows:— Each rigid body, which conformed to the condition that a, ß and £ shall be two pairs of corresponding impulsive and instantaneous screws, will have a different screw f corresponding to a given screw y. Thus, among the various screws in the degraded cylindroid, each will correspond to one rigid body. In general, of course, it would be impossible for £" to be reciprocal to 7. It would be impossible for an impulsive wrench to make a body twist about a screw reciprocal thereto. N evertheless, it seemed certain that, in general, there must be a screw f reciprocal to y. For otherwise, a, ß, 7 should be all parallel to a plane, which, of course, is not generally true. If, however, a, or b, or c were zero, then the body will have no mass; consequently no impulse would be necessary to set it in motion. This clearly is the case when f is reciprocal to 7. We have thus got over the difficulty, f and y are reciprocal, in the case when the rigid body is such that a, or b, or c is zero. 301. Two Geometrical Theorems. The perpendicular from the centre of gravity on any instantaneous screw is parallel to the shortest distance between that instantaneous screw and the corresponding impulsive screw. The perpendicular from the centre of gravity on any instantaneous screw is equal to the product of the pitch of that screw, and the tangent of the angle between it and the corresponding impulsive screw.