A Treatise on the Theory of Screws

394 THE THEORY OF SCREWS. [356, where (11), (12), &c., are ri2 coefficients depending on the distribution of the masses, and the other circumstances of the mass-chain and its constraints. The equations having this form, the necessary one-to-one correspondence is manifestly observed. 357. The principal screw-chains of Inertia. We are now in a position to obtain a result of no little interest. Just as we have two double points in two homographic rows on a line, so we have n double chains in the two homographic chain systems. If we make, in the foregoing equations, — P^l J ^2 “ P^2> we obtain, by elimination of alt ... an, an equation of the nth degree in p. The roots of this equation are n in number, and each root substituted in the equations will enable the co-ordinates of each of the n double screw-chains to be discovered. The mechanical property of these double chains is to be found in the following statement:— If any mass-chain have n degrees of freedom, then in general n screw- chains can always be found (but not more than n), such that if the mass-chain receive an impulsive wrench from any one of these screw-chains, it will immediately commence to move by twisting about the same screw-chain. In the case where the mass-chain reduces to a single rigid body, free or constrained, the n screw-chains to which we have just been conducted reduce to what we have called the n principal screws of inertia. In the case, still more specialized, of a rigid body only free to rotate around a point, the theorem degenerates to the well-known property of the principal axes. We may thus regard the n principal chains now found as the generalization of the familiar property of the principal axes for any system anyhow con- strained. Considerable simplification is introduced into the equations when, instead of choosing the chains of reference arbitrarily, we select the n principal screw-chains for this purpose; we then have the very simple results, ^=(11) a,; 02 = (22) a2; ... 0n = (nn) an. This gives a method of finding the impulsive screw-chain corresponding to any instantaneous screw-chain. It is only necessary to multiply the co- ordinates of the instantaneous screw-chain a1( a2 by the constant factors (11), (12), &c., in order to find the co-ordinates of the impulsive screw-chain. The general type of homography here indicated has to be somewhat specialized for the case of impulsive screw-chains and instantaneous screw- chains. The n double screw-chains are generally quite unconnected—we might, indeed, have exhibited the relation between the two homographic