A Treatise on the Theory of Screws

____________________ 224 ________ ________ _____ ____ _____ THE THEORY OF SCREWS. ______ [214, cylindroid. In like manner let p be the other principal screw of the cylin- droid (3y]r). Then p and a determine the cylindroid (per) which belongs to the system, 6 must lie on the common perpendicular to p and a, and hence the screws of the cylindroid (pa) each intersect 3 at right angles. If 0 is a screw of stationary pitch in a four-system, it can be shown that thiee screws p, a, t not on the same cylindroid can be found in the same system, and such that they intersect 3 at right angles. In this case p, a, t will determine a three-system, every screw of which intersects 3 at right angles. 215. Application to the Two-System. The principles of the last article afford a simple proof of many funda- mental propositions in the theory. We take as the first illustration the well- known fact (§ 76) that if the co-ordinates of a screw satisfy four linear equations then the locus of that screw is a cylindroid. From the general theorem we see by the case of n = 2 that in any two- system a screw of stationary pitch will be intersected at right angles by another screw of the two-system. These two screws may be conveniently taken as the first and third of the canonical co-reciprocal system lying on the axes of x and y. Hence we have as the co-ordinates of a screw of the system 3lt 0, 33, 0, 0, 0. Ihe investigation has thus assumed a very simple form inasmuch as the four linear equations express that of the six co-ordinates of a screw of the system four are actually zero. Let X, p, v be the direction angles of the screw 3 with respect to the associated Cartesian axes then 44), _ (/tø + a) cos A, — dei sin X _ (p0 — a) cos X — dn sin X a ’ —a ’ p __ (Pe 4~ b) cos p — d^ sin p n _ (p6 — b) cos p — d^ sin p o’ -b ' 0 _ (Pe + c)cos ~ d63 sin v _ (pe — c) cos v — de3 sin v c , _ _____ The two last of these equations give cos v = 0; dm = 0. Hence we learn that 3 must intersect the axis of z at right angles. 3 is thus parallel to the plane of xy at a distance dei = dm = z, and accordingly we have the equations (pt, - a) cos X - z sin X = 0, (Pq — b) cos p — z sin p = 0,