A Treatise on the Theory of Screws

IMMM HMM CHAPTER XIV . FREEDOM OF THE THIRD ORDER*. 169. Introduction. The dynamics of a rigid body which has freedom of the third order, possesses a special claim to attention, for, included as a particular case, we have the celebrated problem of the rotation of a rigid body about a fixed point. In the theory of screws the screw system of the third order is characterised by the feature that the reciprocal screw system is also of the third order, and this is a fertile source of interesting theorems. We shall first study the screw system of the third order, and its reciprocal. We shall then show how the instantaneous screw, corresponding to a given impulsive screw, can be determined for a rigid body whose movements are prescribed by any screw system of the third order. We shall also point out the three principal screws of inertia, of which the three principal axes are only special cases, and we shall determine the kinetic energy acquired by a given impulse. Finally, we shall determine the three harmonic screws, and we shall apply these principles to the discussion of the small oscillations of a rigid body about a fixed point under the influence of gravity. A screw system of the first order consists of course of one screw. A screw system of the second order consists of all the screws on a certain ruled surface (the cylindroid). Ascending one step higher, we find that in a screw system of the third order the screws are so numerous that a finite number (three) can be drawn through every point in space. In the screw system of the fourth order a cone of screws can be drawn through every point, while to a screw system of the fifth order belongs a screw of suitable pitch on every straight line in space. 170. Screw System of the Third Order. We shall now consider the collocation of the screws in space which constitute a screw system of the third order. A free rigid body can receive * Transactions of the Royal Irish Academy, Vol. xxv. p. 191 (1871).