A Treatise on the Theory of Screws

[274- 290 THE THEORY OF SCREWS. 275. Geometrical meaning. The nature of the pitch invariant function can be otherwise seen. It is well known that in the composition of two or more twist velocities we „ A2 cos Cj _ „ Pl For, take a ray e, which makes direction-angles X, p, v with a, b, c, then we have cos e, = COS X COS a, + COS p COS + COS V COS C] cos e6 = cos X cos a6 + cos p cos Z>6 + cos v cos c6. Hence .41 cos e, A6 cos e6 _ Æ cos a, 4“ ... 4- —— = cos XX----------- Pi „ A, cos 6, + cos ul------------- Pl v A2 COS 01 "I- COS ------------ If, therefore, we have v A cos ax v A2 cos &i _ Pl Pl then, for every ray, we shall have A2 cos _ — v. Pl It thus follows that the coefficients of a linear function which possesses the property of a pitch invariant must be subjected to three conditions. There are accordingly only three coefficients left disposable in the most general type of linear pitch invariant. Now, fL+gM+hN is a pitch invariant which contains three disposable quantities, f, g, h; it therefore represents the most general form of linear function which possesses the required property. We have thus solved the problem of finding a perfectly general expression for the linear pitch invariant function of the co-ordinates of a screw. It is convenient to take the three fundamental rays as mutually rect- angular ; but it is, of course, easy to show that any linear pitch invariant can be expressed in terms of three pitch invariants unless their determining rays are coplanar. We may express the result thus:—Let L, M, N, 0 be four linear pitch invariants, no three of which have coplanar determining rays. Then it is always possible to find four parameters, X, p, v, p, such that the following equation shall be satisfied identically:— \L + pM + vN + pO = 0.