A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 537 the special case when two reciprocal screw-systems have a screw in common. I had overlooked this exception (§ 96). The existence of n real principal screws of inertia of a rigid body with n degrees of freedom is proved also by Octonions, p, 248, and also the n harmonic screws, p. 248. Joly (C. J.)—The Associative Algebra applicable to Hyperspace. Proceedings of the Royal Irish Academy, 3rd Ser., Vol. v., No. 1, pp. 73-123 (1898). The algebra considered in the present paper is that where units in i2...iH satisfy equations of the type %2 = -1 and i,it + iti, = 0. In this profound memoir there is a discussion of the Theory of Screws in a space of m dimensions. We learn that “when a system compounded from m screws is defined by a linear function (/), the reciprocal system is defined by the negative of the conjugate of that function (-/')■” The canonical representation of a screw in Hyperspace is given and the vector equation to the locus which is the analogue of the cylindroid. The following result, p. 106, is of much interest. “ Thus in spaces of even order, the general displacement of a body may be effected by rotations of definite amounts in a number of definite hyper-perpendicular planes, one determinate point being held fixed■ in spaces of odd order, a translational displacement must be added to the generalized rotation ; but by proper choice of base-point this displacement may be made perpendicular to all the planes of rotation.” This remark is illustrated by the well-known laws of the displacement of a body in two or three dimensions respectively. Joly (C. J.)—Bishop Law’s Mathematical Prize Examination in the University of Dublin, Michaelmas, 1898. Many of Sir William Hamilton’s discoveries in quaternions were first announced in questions which he proposed from time to time at the Law Prize examination. This is, so far as I know, the only examination in which quaternion problems are still habitually proposed. Professor C. J. Joly in the Law Prize paper for 1898 has given the following questions containing applications of Quater- nions to the Theory of Screws. (a) The origin being taken as base-point, let p and Å denote the couple and the force of any wrench, then the transformation AAA contains Poinsot’s theorem of the Central Moment. (6) p = V + x (A, + sX2) x ' Ai 4- SA2 is the vector equation of a ruled surface (the cylindroid) formed by the central axes of wrenches compounded from two given wrenches (/*„ XJ and (/j2, Å2). (c) The form of this equation shows that the locus of the feet of per- pendiculars dropped from an arbitrary point on the generators of a cylindroid is a conic section. (d) If Å) is any wrench compounded from three given wrenches A,), Å.,), and (p3, Xj, the couple of this wrench is a determinate linear vector function of the force, or p <£A, and the function </> adequately defines this ‘ three- system’ of wrenches.