A Treatise on the Theory of Screws

 346 THE THEORY OF SCREWS. [318, If the former had been applied to J/,' it would have generated about a (§ 280) a twist velocity represented by sin — <f>) 1 cos (0a.) sin(0 — </>) Mi pa If the latter had been applied to the body Mf, it would have generated a twist velocity about the same screw, a, equal to ,„ sin (0 — \|r) 1 cos (fa) * sin (0 — f) Mt' pa Suppose that these two twist velocities are equal, it is plain that the original wrench on f would, if it had been applied to the composite rigid body, produce a twisting motion about a. The condition is sin (i/r — </>) __ sin (0 — i/r) Mf cos (fa) M2 cos (0d) ’ We thus obtain tan f in terms of Mf : M.f. As the structure of the composite body changes by alterations of the relative values of p, and p2, so will f move over the various screws of the cylindroid (0, </>). This result shows that, if three screws, a, ß, y be given, then the possible impulsive screws, y, %, if which shall respectively correspond to a, ß, y in a given system of the third order P, are uniquely determined. For, suppose that a second group of screws, y', g, If, could also be deter- mined which fulfilled the same property. We have shown how another rigid body could be constructed so that another screw, could be found on the cylindroid (y, y'), such that an impulse thereon given would make the body twist about a. It is plain that, for this body also, the impulsive wrench, corresponding to ß, would be some screw on the cylindroid (£, £'). But all screws on this cylindroid belong to the system P. In like manner, the instantaneous screw y would correspond for the composite body to some screw on the cylindroid (f if). Hence it follows that, for each different value of the ratio p, : p.2, we would have a different set of impulsive screws for the instantaneous screws a, ß, y. We thus find that, if there were more than one set of such impulsive screws to be found in the system P, there would be an infinite number of such sets. But we have already shown that the number of sets must be finite. Hence there can only be one set of screws, y, if in the system P, which could be impulsive screws corresponding to the instantaneous screws, a, ß, y. We are thus led to the following important theorem, which will be otherwise proved in the next chapter. Given any two systems of screws of the third order. It is generally pos- sible, in one way, but only in one, to design, and place in a particular position a