A Treatise on the Theory of Screws

291 EMANANTS AND PITCH INVARIANTS. cos «!+... + 06 cos a6 — 0, 0} cos . + 06 cos b6 — 0, 0! cos cx + ... + 06 cos c6 = 0. The screws which satisfy these conditions must all be perpendicular to the 19—2 276] discover the direction of the resultant screw and the magnitude of the resultant twist velocity by proceeding as if the twist velocities were vectors. Neither the pitches of the component screws nor their situations affect the magnitude of the resultant twist velocity or the direction of the resultant screw. This principle is, of course, an immediate consequence of the law of composition of twist velocities by the cylindroid. Let any ray a make an angle X with the ray 0, and angles alt ... a6 with the six screws of reference. The twist velocity 0 on 0 if resolved on a has a component 0 cos X. This must be equal to the sum of the several com- ponents 0lt 02, ... resolved on a; whence we have 0 cos X — 01 cos ßj + ... + 0(5 cos a6. If we make 0 = unity, we obtain cos X = 01 cos a, + . •. + 06 cos ae. This gives a geometrical meaning to the pitch invariant. It is simply the cosine of the angle between the screws 0 and a. As, of course, the pitch is not involved in the notion of this angle, it is, indeed, obvious that the expression for any function of the angle must be a pitch invariant. We now see the meaning of the equation obtained by equating the pitch invariant to zero. If we make 0} cos . + 06 cos ae = 0 it follows that a and 0 must be at right angles. The equation therefore signifies the locus of all the screws that are at right angles to a. The two equations 02 cos cij + ... + 0e cos a6 = 0, 01 cos . + 06 cos b$ — 0, denote the screws perpendicular to the two directions of a and ß. In other words, these two equations define all the screws perpendicular to a given plane. 276. Screws at infinity. Let us now take the case where a, ß, 7 are three rectangular screws, and examine the conditions satisfied by 0ly ..., 06 when subjected to the three following equations: