A Treatise on the Theory of Screws

______________ ______ ___________ 186 .... _____ _______________ _____________ _________________ THE THEOBY OF SCREWS. __________ [184- the equation becomes 0 = P«.—p , — R — p cos (a/3), + Q' —p cos (ay) R' - p cos {aß), Pfi-p , - P’ - p cos {ßy) - Q'-pcos{ay), +P'-p cos{ßy), Py — P By expanding this as a cubic for p we see that the coefficient of p2 divided by that of p3 with its sign changed is _________________________ffqsin2(/3y) +pß sin2 (ya) +py sin2 (aß) 4 sinI[(ßy) + (ya) + (aß)] sin j [(/3-y) + (ya) - (aß)]sin|[(/3y) - (ya) + (aß)]sin j[ - (ßy) + (ya) + (aß)] ' This is accordingly the constant sum of the three pitches of the screws of the system which can be drawn through any point. 185. Equilibrium of Four Forces applied to a Rigid Body. If the body be free, the four forces must be four wrenches on screws of zero pitch which are members of a screw system of the third order. The forces must therefore be generators of a hyperboloid, all belonging to the same system (§ 132). Three of the forces, P, Q, R, being given in position, £ must then be a generator of the hyperboloid determined by P, Q, R. This proof of a well-known theorem (due to Möbius) is given to show the facility with which such results flow from the Theory of Screws. Suppose, however, that the body have only freedom of the fifth order, we shall find that somewhat more latitude exists with reference to the choice of »S'. Let X be the screw reciprocal to the screw system by which the freedom is defined. Then for equilibrium it will only be necessary that $ belong to the system of the fourth order defined by the four screws P, Q, R, X. A cone of screws can be drawn through every point in space belonging to this system, and on that cone one screw of zero pitch can always be found (§ 123). Hence one line can be drawn through every point in space along which S might act. If the body have freedom of the fourth order, the latitude in the choice of S is still greater. Let X., be two screws reciprocal to the system, then 6' is only restrained by the condition that it belong to the screw system of the fifth order defined by the screws __________________ ________________________________ ______________________