A Treatise on the Theory of Screws

286] DEVELOPMENTS OF THE DYNAMICAL THEORY. 305 Multiplying similarly by/jj/Si, ...pnßn, and adding, H u> qß ~~ • Eliminating H, we find WaaS’øi) — U<iß^a-q = 0. We may also prove this formula by physical consideration. Let a, ß be the two screws which correspond, as instantaneous screws, to t) and as impulsive screws. Let us take on the cylindroid a, ß, a screw 9, which is conjugate to a with respect to inertia (§ 81). Then, by known principles, the screw 9 so defined must be reciprocal to tj. Hence p1t)191 +...+pnt)n9n = 0. As, however, a and 9 are conjugate, we have u^a, 9X + ... + Mn2a„ 9n = 0; also, since 9 is co-cylindroidal with a and ß, there must be relations of the kind 91 = >a1 +pßj-, ... 9n = Xan +pßn. Substituting these in the two previous equations, we get 4- p^^ß = 0 ; Xuaa + puaß = 0 ; whence, as before, Uaat^ßv ~ Uapttfay = 0. 286. Twist Velocity acquired by an Impulse. From the fact that the twist velocity ä acquired by a free body in consequence of an impulsive wrench of unit intensity on a screw y is expressed (§ 280) by the equation cos (at)) a —-------- Pa we see that the second of the two formulae of § 281 may be expressed thus:— ß^ßn = The proof thus given of this expression has assumed that the body is quite free. It is however a remarkable fact that this formula holds good whatever be the constraints to which the body is submitted. If the body receive the unit impulsive wrench on a screw t), the body will commence to twist about b. 20