A Treatise on the Theory of Screws

saanæ 252 THE THEORY OF SCREWS. [233- 233. Analytical Method. A quiescent rigid body which has freedom of the fifth order receives an impulsive wrench on a screw y: it is required to determine the instantaneous screw 6, about which the body will commence to twist. Let X be the screw reciprocal to the freedom, and let the co-ordinates be referred to the absolute principal screws of inertia. The given wrench com- pounded with a certain wrench on A, must constitute the wrench which, if the body were free, would make it twist about 0, whence we deduce the six equations (h being an unknown quantity) hpßs — V Ps + X "X6. Multiplying the first of these equations by Xj, the second by X2) &c., adding the six equations thus produced, and remembering that 0 and X are reciprocal, we deduce = 0. This equation determines X'" the impulsive intensity of the reaction of the constraints. The co-ordinates of the required screw 0 are, therefore, proportional to the six quantities Pi ’ ■“ 234. Principal Screws of Inertia. We can now determine the co-ordinates of the five principal screws of inertia; for if £ be a principal screw of inertia, then in general whence with similar values for £2, ... £6. Substituting these values in the equation PiK^i + P^-2 + PiK^i +pi^i+p5K^ + Ps\£<s = 0, t"' and making -y = x, we have for x the equation Pl\l PP^i , Ps^s2 Pi^-42 . Ppl-P Pe\>" -------j-------1--------(----------------1------_ y. Pi — X p.2- X Pi - x pl- X ps- X Pi — X This equation is of the fifth degree, corresponding to the five principal screws of inertia. If x' denote one of the roots of the equation, then the corresponding principal screw of inertia has co-ordinates proportional to Xi Xg A/g ^4 ^*5 A-g Pl - x! ’ p2 - X ’ p3- x” Pl- x' ’ p5-x” Pi- x''