A Treatise on the Theory of Screws

305] THE GEOMETRICAL THEORY. 325 304. An Important Exception. If pa = 0, then (ay) is 90°, and consequently pa tan (ay) is indefinite. If, therefore, the pitch of the instantaneous screw be zero, then we are no longer entitled to locate the centre of gravity in a certain ray. All we know is that it lies in the plane through a perpendicular to y. In general the knowledge of the impulsive screw corresponding to a given instantaneous screw implies five data, yet this ceases to be the case if pa is zero, for as y must then be perpendicular to a there are really only four independent data given when y is given. We have, therefore, in this case one element the less towards the determination of the rigid body. 305. Two Pairs of Impulsive and Instantaneous Screws. Let us next suppose that we are given a second pair of corresponding impulsive and instantaneous screws. We shall examine how much further we are enabled to proceed by the help of this additional information towards the complete determination of the rigid body in its abstract form. Any data in excess of nine, if not actually impossible, would be superfluous. If, therefore, we are given a second pair of impulsive and instantaneous screws, the five data which they bring cannot be wholly independent of the five data brought by the preceding pair. It is therefore plain that the quartet of screws forming two pairs of corresponding impulsive screws and instantaneous screws cannot be chosen arbitrarily. They must submit to at least one purely geometrical condition, so that the number of data independent of each other shall not exceed nine. It is, however, not so obvious, though it is certainly true, as we found in § 281, that the two pairs of screws must conform not merely to one, but to no less than two geometrical conditions. In fact, if y, % be two impulsive screws, and if a, ß be the two corresponding instantaneous screws, then, when the body acted upon is perfectly free, the following two formulæ must be satisfied: P« -_______ cos (ay) cos (ߣ) a ’ c“ w+SJW “* (“f >' We can enunciate two geometrical properties of the two pairs of screws, which are equivalent to the conditions expressed by these equations. In the first place, each of the pairs of screws determines a diameter of the momental ellipsoid. The fact that the two diameters, so found, must intersect each other, is obviously one geometrical condition imposed on the system a, y and ß, %.