A Treatise on the Theory of Screws

214] FREEDOM OF THE FOURTH ORDER. 221 only movement which the body can receive, so as to fulfil the prescribed conditions, is a twist about the screw X. For X is then reciprocal to and therefore a body twisted about X will do no work against forces directed along u41;... As. From the theory of reciprocal screws it follows that a body rotated around any of the lines Alt... As will not do work against nor receive energy from a wrench on X. In the particular case, where A1;... As have a common transversal, then X is that transversal, and its pitch is zero. In this case it is sufficiently obvious that forces on Alt ...As cannot disturb the equilibrium of a body only free to rotate about X. 214. Screws of Stationary Pitch. We begin by investigating the screws in an n-system of which the pitch is stationary in the sense employed in the Theory of Maximum and Minimum. We take the case of n = 4. The co-ordinates 01;... 36 of the screws of a four-system have to satisfy the two linear equations defining the system. We may write these equations in the form Ai + ... + AsOfs — 0, 5^+...+ 5,^ = 0. The screws of reference being co-reciprocal, we have for the pitch pe the equation 2pi0ia - Rp0 = 0, where R is the homogeneous function of the second degree in the co- ordinates which is replaced by unity (§ 35) in the formulae after differ- entiation. If the pitch be stationary, then by the ordinary rules of the differential calculus (§ 38), 8^ + ... + (2^-M = 0. As however 6 belongs to the four-system, the variations of its co-ordinates must satisfy the two conditions ^M+.-+^6 = 0> B^ex + ... + w=o. Following the usual process we multiply the first of these equations by some indeterminate multiplier Å, the second by another quantity p, and then