A Treatise on the Theory of Screws

334] VARIOUS EXERCISES. 363 Hence from knowing the two cylindroids eight of the co-ordinates of the rigid body are uniquely fixed while the ninth remains quite indeter- minate. Every value for p32 will give one of the family of rigid bodies for which the desired condition is fulfilled. We have already deduced geometrically (§ 306) the relations of these rigid bodies. We now obtain the same results otherwise. The rnomental ellipsoid around the centre of gravity has as its equation - ®0)2 p/ + (y - y„)2 p2 + (z - ^0)2 p32 - 2 (y - y0) (z - z„) I2 - 2 (z - z0) (« - «„) l22 - 2 (a; - «„) (y - y0~) la2 - (yxa - xy0)2 - (zy„ - yztl)2 - (a:z0 - zxtf = k1. This may be written in the form where p32 does not enter into R. As p32 varies this equation represents a family of quadrics which have contact along the section of R = 0 by the plane z — z„ - 0. This proves that a plane through the common centre of gravity and parallel to the principal plane of the cylindroid U passes through the conic along which the rnomental ellipsoids of all the different possible bodies have contact. All these quadrics touch a common cylinder along this conic. The infinite point on the axis of this cylinder is the pole of the plane z — z3 = 0 for each quadric. Every chord parallel to the axis of the cylinder passes through this pole and is divided harmonically by the pole and the plane z — z3 = 0. As the pole is at infinity it follows that in every quadric of the system a chord parallel to the axis of the cylinder is bisected by z — z3. Hence a diameter parallel to the axis of the cylinder is conjugate to the plane z — ztt in every one of the quadrics. Thus by a different method we arrive at the theorems of § 306. 334. The Double Correspondents on Two Cylindroids. Referring to the remarkable homography between the impulsive screws on one cylindroid V and the corresponding instantaneous screws on another cylindroid R we have now another point to notice. If the screws on U were the impulsive screws, while those on V were the instantaneous screws, there would also have been a unique homography, the rigid bodies involved being generally distinct. But of course these homographies are in general quite different, that is to say, if J. be a screw in U the instantaneous cylindroid, and B be its corre- spondent in V the impulsive cylindroid, it will not in general be true that if