A Treatise on the Theory of Screws

232 THE THEORY OF SCREWS. [221- screw corresponding to those of a given impulsive screw can be deduced from Eulers theorem (§94). If a body receive an impulsive wrench on a screw while the body is constrained to twist about a screw 0, then we have seen in § 91 that the kinetic energy acquired is proportional to If 6i> 0-a, Ø3, be the co-ordinates of 0 referred to the four principal screws of inertia belonging to the screw system of the fourth order then (§§ 95, 97) = (pyA + p.1V202 + p.3V.30.3 + piVi0^, ui = uffi + u/0.? + ui3i + ui3i. Hence we have to determine the four independent variables 0lt 02, 0S, 04, so that (PlVlOl +P2V2O2 + WÄ + jwA)2 ui3i + u.20.;- + uiøi + u4-02 ’ shall be stationary. This is easily seen to be the case when 04, 02, 03, 04 are respectively proportional to Pl P2 P-3 Pi ct/l CV3 164 These are accordingly, as we already know (§ 97), the co-ordinates of the screw about which the body will commence to twist after it has received an impulsive wrench on y. This method might of course be applied to any order of freedom. 222. General Remarks. It has been shown in §80 how the co-ordinates of the instantaneous screw corresponding to a given impulsive screw can be determined when the ligid body is perfectly free. It will be observed that the connexion between the two screws depends only upon the three principal axes through the centre of inertia, and the radii of gyration about these axes. We may express this result more compactly by the well-known conception of the momenta! ellipsoid. The centre of the inomental ellipsoid is at the centre of. inertia of the rigid body, the directions of the principal axes of the ellipsoid are the same as the principal axes of inertia, and the lengths of the axes of the ellipsoid are inversely proportional to the corresponding radii of gyration. When, therefore, the impulsive screw is given, the momenta! ellipsoid alone must be capable of determining the corresponding instantaneous screw. A family of rigid bodies may be conceived which have a common