A Treatise on the Theory of Screws

324 THE THEORY OF SCREWS. [303- We can now determine the value of where pa is the radius of gyration about an axis parallel to a through the centre of gravity. For the kinetic energy is obviously JJ/d2 (pa2+po2 + aa2). By equating the two expressions we have But when a and r) are known the three terms on the right-hand side of this equation are determined. Thus we learn the radius of gyration on the diameter parallel to a. It remains to show how a certain straight line in the plane which is conjugate to this diameter in the momental ellipsoid is also determined. Let a screw 0, of zero pitch, be placed on that known diameter of the momental ellipsoid which is parallel to a. Draw a cylindroid through the two screws 0 and y. Let <f> be the other screw of the zero pitch, which will in general be found on the same cylindroid. We could replace the original impulsive wrench on y by its two com- ponent wrenches on any two screws of the cylindroid. We choose for this purpose the two screws of zero pitch 0 and </>. Thus we replace the wrench on i) by two forces, whose joint effect is identical with the effect that would have been produced by the wrench on r/. As to the force along the line 0 it is, from the nature of the con- struction, directed through the centre of gravity. Such an impulsive force would produce a velocity of translation, but it could have no effect in pro- ducing a rotation. The rotatory part of the initial twist velocity must there- fore be solely the result of the impulsive force on <f>. But when an impulsive force is applied to a quiescent rigid body we know, from Poinsot’s theorem, that the rotatory part of the instantaneous movement must be about an axis parallel to the direction which is conjugate in the momental ellipsoid to the plane which contains both the centre of gravity and the impulsive force. It follows that the ray must be situated in that plane which is conjugate in the momental ellipsoid to the diameter parallel to a. But, as we have already seen, the position of </> is completely defined on the known cylindroid on which it lies. We have thus obtained a fixed ray in the conjugate plane to a known diameter of the momental ellipsoid. The three statements at the beginning of this article have therefore been established. We have, accordingly, ascertained five distinct geometrical data towards the nine which are necessary for the complete specification of the rigid body. These five data are inferred solely from our knowledge of a single pair of corresponding impulsive and instantaneous screws.