A Treatise on the Theory of Screws

74 THE THEORY OF SCREWS. [84, In like manner if ß, 7 be two other screws of the three-system, + pW&i +... + pe2ßA = xp&e, + . + xpAoe, Pi^idi +p32y30., + ... + /V?<A = «P17J0, 4 xp2y.ß2... + xp^Ø*. But as 0 belongs to the three-system its co-ordinates must satisfy three linear equations. These we may take to be FA+ FA+ ... + FA = 0, + G202 + ... 4 Ge06 — 0, H10l + H202 + ... + Hli0li = G We have thus six linear equations in the co-ordinates of 0. We can therefore eliminate those co-ordinates, thus obtaining a determinantal equation which gives a cubic for x. The three roots of this cubic will give accordingly three screws in the three-system which possess the required property. Ihus we demonstrate thnt in any three-system there ai*6 three principal screws of inertia, and a precisely similar proof for each of the six values of n establishes by induction the important theorem that there are n principal screws of inertia in the screw system of the nth order. It is shown in § 86 that all the roots are real. We shall now prove that the Principal Screws of Inertia are co-reciprocal. Let 0 and $ be two such screws, corresponding to different roots x', x" of the equation in x. Then we have 0i =-~y^, Pl-X 0^_y^_ 0 _yK p2-^ p6-x'' Let p be the screw of the reciprocal system on which the impulsive wrench is generated by the impulse given on </>. Then </>■ Pi - x Pi , , ypo , •••</>«-------- x pr,~ x As p, is reciprocal to 0 titid X is reciprocal to cf>} wg have Pi\Pi . P2KP2 PhXhPh i • f i • • • r"-• = U — x p2 — x pa — x ’ Pl\pl . Pn\lP2 , »6X6/ie Pi-x"^p2-x"+- Subtracting these equations and discarding the factor x - x", we get ----__________ + (Pi-x^p.-x") (pi-x)(p2-x")^ "•