A Treatise on the Theory of Screws

300 [282, THE THEORY OF SCREWS. The formula becomes, when is very large, P°- Pß i , ----7—\ * Pa cos (at). cos (a??) " cos (fig) v s' In this case as the twist about a is merely a translation, we must have cos ap = 1, so that 1 Pß , ~ ^ßv is to vanish, but this cannot be secured by making pß zero, because that cannot happen without cos (ß’p) being zero, except the pitch of f be infinite (§ 280) which is the case already excluded. It is therefore necessary that cos (af) be zero, but this requires that a and f be reciprocal, i.e. that = 0. Let us now suppose that we try to satisfy the original equations by making =0, pß = 0. Here again we find that pß = 0 entails cos (/3£) zero, except />f = oc. This in general makes infinite so that the equation is not satisfied. If a. and £ were at right angles then no doubt the equation would be satisfied, but then is zero. We thus see that notwithstanding the special form of the fundamental equation (§ 281) it implies no departure from the complete generality of the principle that whenever is zero then must be also zero. 283. A Fundamental Theorem. Let us suppose that a rigid body is either entirely free or constrained in any manner whatever. Let p be an impulsive screw whose pitch pv is not infinite. Let p'" be the intensity of an impulsive wrench on that screw, it being understood that p'" is to be neither zero nor infinity. Let a be the instantaneous screw about which the body, having been previously quiescent, will commence to twist with an instantaneous twist velocity a. It is also supposed that pa is neither zero nor infinity. Let f be the impulsive screw similarly related to ß, and let the affiliated symbols have the corresponding significations and limitations. Let £ be the impulsive screw similarly related to y, and let the affiliated symbols have the corresponding significations and limitations. The instantaneous movement of the body must necessarily be the same as if it had been quite free and had received in addition to the impulsive wrench of intensity p'" on the screw t,, an impulsive wrench of intensity p" situated on some screw p belonging to the system of screws reciprocal to the freedom of the body. Let these two wrenches compound into a single wrench of intensity co'" on a screw co.