A Treatise on the Theory of Screws

84] the principal screws of inertia. 73 so that every pair of them are conjugate screws of inertia (§ 81). Let /?,,&('„ be (6 — /l) screws defining the reciprocal screw system. Let be any screw belonging to P. Then in the choice of we have n — 1 arbitrary quantities. Let P be any impulsive screw corresponding to yl, as an instan- taneous screw. Choose A., reciprocal to ZX1 ... 56-»> then and A2 are conjugate screws, and in the choice of the latter we have n — 2 arbitrary quantities. Let Z2 be any impulsive screw corresponding to A2 as an instan- taneous screw. Choose A3 reciprocal to P, I2, Ip, ... lp-n, and proceed thus until JLn has been attained, then each pair of the group .J1; &c. An are conjugate screws of inertia. The number of quantities which remain arbitrary in the choice of such a group amount to „ , n(n— 1) 11 — 1 + TO — 2 + ... + 1 =-a--, or exactly half the total number of arbitrary constants disposable in the selection of any n screws from a system of the nth order. 84. Principal Screws of Inertia. We have now to prove the important theorem in Dynamics which affirms the existence of n principal Screws of Inertia in a rigid body with n degrees of freedom. The proof that we shall give is, for the sake of convenience, enunciated with respect to the freedom of the third order, but the same method applies to each of the other degrees of freedom. Let 0 be one of the principal screws of inertia, then an impulsive wrench on 0 must make the body commence to twist about 0. In the most general case when the body is submitted to constraint, the impulsive wrench on 0 will of course be compounded with the reaction on some screw Å. of the reciprocal system. The result will be to produce the impulsive wrench which would, if the body had been free, have generated an instantaneous twist velocity about 0. We thus have the following equations (§ 80) where x and y are unknown: p10l = x02 + yXi, p20'i — x02 + yX2, pPP = »P, + yK- Let a be one of the screws of the three-system in question. Then since X must be reciprocal to a we have by multiplying these equations respectively by p^i,... p^6 anå adding, + •••+ PfaO» = xpiuA 4- xp2a202 + ... + xpsae0e.