A Treatise on the Theory of Screws

329J VARIOUS EXERCISES. 357 327. Another proof of article 303. As an illustration of the formulae just given we may verify a theorem of § 303 showing that when we know the instantaneous screw corresponding to a given impulsive screw, then a ray along which the centre of gravity must lie is determined. For subtracting the second equation from the first and repeating the process with each of the other pairs, we have e (vi + Vz) = (ai - «2) + 2^> (“a + «4) - 2y0 (a5 + a„), e (Vs + Vt) = (a3 — a4) + 2a?0 (a5 + a«) — 2^0 (a, + a2), e + Vs) = 2c (a6 - a„) + 2y0 (ax + a2) - 2«0 (a3 + a4). Eliminating e we have two linear equations in thus proving the theorem. If we multiply these equations by ai + a2, a3 + a4, a6 + a6 respectively and add, we obtain e cos (ay) = 2pa, thus giving a value for e. 328. A more general Theorem. If an instantaneous screw be given while nothing further is known as to the rigid body except that the impulsive screw is parallel to a given plane A , then the locus of the centre of gravity is a determinate plane. Let X, p, v be the direction cosines of a normal to A, then X (vi + Vn) + P‘ (Vs + Vt) + v (Vs + Vs) = 0, whence by substitution from the equations of the last Article we have a linear equation for x0, y„, z0. 329. Two Three-Systems. We give here another demonstration of the important theorem of § 318, which states that when two arbitrary three-systems Ü and V are given, it is in general possible to design and place a rigid body in one way but only in one way, such that an impulsive wrench delivered on any screw y of V shall make the body commence to move by twisting about some screw a of U. Let the three principal screws of the system U have pitches a, b, c and take on the same three axes screws with the pitches — a, — b, — c respectively. These six screws lying in pairs with equal and opposite pitches form the canonical co-reciprocals to be used.