A Treatise on the Theory of Screws

10] THE CYLINDROID. 17 body as two or more given twists (or wrenches) about two or more given screws. It will be found to be a fundamental point of the present theory that the rules for the composition of twists and of wrenches are identical*. 10. The Virtual Coefficient. Suppose a rigid body be acted upon by a wrench on a screw ß, of which the intensity is ß". Let the body receive a twist of small amplitude a' around a screw a. It is proposed to find an expression for the energy required to effect the displacement. Let d be the shortest distance between a and ß, and let 0 be the angle between a and ß. Take a as the axis of x, the common perpendicular to a and ß as the axis of z, and a line perpendicular to x and z for y. If we resolve the wrench on ß into forces X, Y, Z, parallel to the axes, and couples of moments L, M, N, in planes perpendicular to the axes we shall have X = ß" cos 0; Y = ß" sin 9 ; Z = 0; L = ß”pß cos 9 — ß"d sin 9; M = ß”pß sin 9 + ß"d cos Q ; 2V = 0. We thus replace the given wrench by four wrenches, viz., two forces and two couples, and we replace the given twist by two twists, viz., one rotation and one translation. The work done by the given twist against the given wrench must equal the sum of the eight quantities of work done by each of the two component twists against each of the four component wrenches. Six of these quantities are zero. In fact a rotation through the angle a around the axis of x can do work only against L, the amount being aß" (pø cos 9— d sin 9). The translation pad parallel to the axis of x can do work only against X, the amount being a'ß"pa cos 0. Thus the total quantity of work done is aß” {(Pa+Pß) cos 0 - d sin 0}. The expression | [(pa + Pß) cos 0 - d sin 0] is of great importance in the present theory]*. It is called the virtual * That the analogy between the composition of forces and of rotations can be deduced from the general principle of virtual velocities has been proved by Rodrigues (Liouville's Journal, t. 5, 1840, p. 436). + The theory of screws has many points of connexion with certain geometrical researches on the linear complex, by Plücker and Klein. Thus the latter has shown (Mathematische Annalen, Band ii., p. 368 (1869)), that if pa and pß be each the “Hauptparameter” of a linear complex, and if +pß) cos O - d sin 0=0, where d and O relate to the principal axes of the complexes, then the two complexes possess a special relation and are said to be in “ involution.’ B. 2