A Treatise on the Theory of Screws

420 THE THEORY OF SCREWS. [383- applied on the screw corresponding to R, to prohibit the body from changing its instantaneous screw. Let 0' be the pole of the axis of inertia, then, if IA be a chord drawn through O', the points I and A correspond to a pair of conjugate screws of inertia (§ 135). It further appears that A is the instantaneous screw corre- sponding to an impulsive wrench on R (§ 140). Therefore the effect of the wrench on R when applied to control the body twisting about I is to com- pound its movement with a nascent twist velocity about A. Therefore A must be the accelerating screw corresponding to I. We thus see that— Of two conjugate screws of inertia, for a rigid body with two degrees of freedom, either is the accelerator for a body animated by a twist velocity about the other. 384. Calculation of T. In the case of freedom of the second order we are enabled to obtain the form of T, from the fact that the emanant vanishes, that is, Å dTi dT eidø/ + e'2dø/~Q' If we assume that T is a homogeneous function of the second degree in 0, and 0.2, the solution of this equation must be T = L0/ + 280,0, + MØ/ + H (0/0, - 0/0/y + (0/0, - 0/ 0/) (A 0, + B0,\ in which L, 8, M, H, A are constants. If we further suppose that 0/ and 0/ are so small that their squares may be neglected, then the term multiplied by H may be discarded, and we have T = L0/ + 280,0, + M0/ + (0/0, - 0/0,) (A 0, + B0.j), whence dT . . (rr . . -r , = + 0, (A0, + B0j); = - 0, (A0, + B0/). CLU] (tU 2 Thus, for the co-ordinates of the restraining screw, supposing the screws of reference to be reciprocal, we have - S 3F - + t + “■> ’ - S' U ' - - + jPl a<72 p2 from which it is evident that PiVi'0, + p&h'Ø, = 0, which is, of course, merely expressing the fact that y and 0 are reciprocal. 385. Another method. It may be useful to show how the form of T, just obtained, can be derived from direct calculation. I merely set down here the steps of the work and the final result.