A Treatise on the Theory of Screws

■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■i 114 THE THEORY OF SCREWS. (126— 126. Reaction of Constraints. Whatever the constraints may be, their reaction produces an impulsive wrench R} upon the body at the moment of action of the impulsive wrench X2. The two wrenches X, and R, compound into a third wrench F,. If the body were free, Y2 is the impulsive wrench to which the instantaneous screw A1 would correspond. Since X,, X.,, X3 are co-cylindroidal, Alt A2, A3 must be co-cylindroidal, and therefore also must be Ylt Y2, Y3. The nine wrenches Xy, X2, X3, Rlt R2, Rs, - Ylt — Y2, — Ya must equilibrate; but if Xj, X2, X3 equilibrate, then the twist velocities about A2, A2> 43 must neutralize, and therefore the wrenches about Yz, Y2, F3 must equilibrate. Hence Rlt R2, R> equilibrate, and are therefore co-cylindroidal. Following the same line of proof used in the last section, we can show that If impulsive wrenches on any four co-cylindroidal screws act upon a partially free rigid body, the four corresponding initial reactions of the constraints also constitute wrenches about four co-cylindroidal screws; and, further, the anharmonic ratios of the two groups of four screws are equal. 127. Principal Screws of Inertia. If a quiescent body with freedom of the second order receive impulsive wrenches on three screws X2, X2, X3 on the cylindroid which expresses the freedom, and if the corresponding instantaneous screws on the same cylin- droid be t!,, A2, A:!, then the relation between any other impulsive screw X on the cylindroid and the corresponding instantaneous screw A is completely defined by the condition that the anharmonic ratio of X, Xlt X2, X3 is equal to the anharmonic ratio of A, At, A2, A3. If three rays parallel to Xlt X2, X3 be drawn from a point, and from the same point three rays parallel to AJt A2, A3, then, all six rays being in the same plane, it is well known that the problem to determine a ray Z such that the anharmonic ratio of Z, AIt As, A3 is equal to that of Z, Xy, X2, X3, admits of two solutions. There are, therefore, two screws on a cylindroid such that an impulsive wrench on one of these screws will cause the body to commence to twist about the same screw. We have thus arrived by a special process at the two principal screws of inertia possessed by a body which has freedom of the second order. This is, of course, a particular case of the general theorem of § 78. We shall show in the next section how these screws can be determined in another manner. 128. The Ellipse of Inertia. We have seen (§ 89) that a linear parameter wo may be conceived appro- priate to any screw a of a system, so that when the body is twisting about