A Treatise on the Theory of Screws

APPENDIX I. 487 But n - 2 linear equations in an n-system determine a cylindroid and hence we see that all the screws on this cylindroid will be principal Screws of Inertia. In like manner if there be k repeated roots, i.e. if Wj2 _ w28 Pi P-i Pk' then a,, ... ak are arbitrary but aj.+1, ... an must be each zero. We have thus n^k linear equations in the co-ordinations. They must also satisfy 6 - n equations because they belong to the «.-system and therefore they satisfy in all 6 — n + n — k=& — k equations, whence we deduce that If there be k repeated roots in the determinantal equation of § 86 then to those roots corresponds a k-system of screws each one of which is a principal screw of inertia and there are besides n — k additional principal Screws of Inertia. So far as the cases of n = 2 and n = 3 are concerned the plane representations of Chaps. XII. and XV. render a complete account of the matter. Let 0 (Fig. 10) be the pole of the axis of pitch, § 58, then 0 may lie either inside or outside the circle whose points represent the screws on the cylindroid. Let O' (Fig. 22) be the pole of the axis of inertia, § 140, then O' must lie inside the circle, for otherwise the polar of O' would meet the circle, i.e. there would be one or two real screws about which the body could twist with a finite velocity but with zero kinetic energy. We have seen that the two Principal Screws of Inertia are the points in which the chord 00' cuts the circle. If O' could be on the circle or outside the circle then we might have the two principal Screws of Inertia coalescing, or we might have them both imaginary. As however O' must be within the circle it is generally necessary that the two principal Screws of Inertia shall be both real and distinct. But the points 0 and O' might have coincided. In this case every chord through 0 would have principal Screws of Inertia at its extremities. Thus every point on the circle is in this case a principal Screw of Inertia. We thus see that with Freedom of the second order there are only two possible cases. Either every screw on the cylindroid is a principal Screw of Inertia or there are neither more nor fewer than two such screws, and both real. If a and ß be any two screws on the cylindroid then the conditions that all the screws are Principal Screws of Inertia are PaU2aß = ^aßUa2t Pß'llfß = -0aßUß2' With any rigid body in any position we can arrange any number of cylindroids