A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 533 Cayley (A.)—Non-Eudidian Geometry. Transactions of the Cambridge Philo- sophical Society, Vol. xv., pp. 37-61 (1894). Read Jan. 27, 1890. See also Collected Papers, Vol. xiii., p. 480. This is perhaps the best paper in the English language from which to obtain a general view of the Non-Euclidian Geometry. The development is here conducted mainly along geometrical lines. On this account a study of this paper is specially recommended in connection with Chap. xxvi. of the present volume. Budde (E.)—Allgemeine Mechanik der Punkte und starren Systeme. 2 vols. 8vo. Berlin, 1891. This comprehensive work may be cited in illustration of progress made in the use of the Theory of Screws in advanced text-books of Dynamics in Germany. There is an excellent account of the theory of the cylindroid in Vol. II., pp. 596—603. The only exception, and it is a very small one, which I feel inclined to take to this part of Professor Budde’s work is that he speaks of the composition of Screws. It seems to me better to preserve the notion of a screw as simply a geometrical entity and to speak of the composition rather of twists or of wrenches on the screws than of composition of the screws themselves. Vol. II. pp. 639-644 gives an account of the fundamental parts of the theory of reciprocal screw systems. The geometrical construction for the cone of screws which can be drawn through any point reciprocal to a cylindroid (§ 22), and which was originally given in the Theory of Screws, 1876, p. 23, has been here reproduced. A good account is also given, Vol. n., pp. 905-908, of the geometrical theory of the restraints of the most general type. This subject is developed both by the elegant methods of Mannheim and also by those of the Theory of Screws. Routh (E. J.)—Treatise on Analytical Statics, Vol. i., 2nd Edition, 1896. In this standard work several of the fundamental Theorems of the Theory of Screws will be found. See pp. 202-208. Klein (F.)— Nicht-Eudidische Geometrie: Vorlesung. 1889-90. Ausgearbeitet von Fr. Schilling. Göttingen, 1893. This is a lithographed record of Klein’s lectures. It is invaluable to any one who desires to become acquainted with the further developments of that remark- able Theory which is of such great importance in the subject of this volume as in so many other departments of Mathematics. The bearing of the Theory of Screws in its relation to the Non-Euclidian geometry is discussed by the author. Burnside (W.)—On the Kinematics of Non-Euclidian Space. London Math. Soc. Proceedings, xxvi. 33-56, Nov. 1894. The paper consists of a number of applications of a construction for the resultant of two displacements (or motions), the construction being formally independent of the nature of the space, Euclidian, elliptic or hyperbolic, in which the motions are regarded as taking place. § II. of the paper gives the application to elliptic space. The main point in this case is to deduce' synthetically, from the construction, the existence of finite motions winch correspond to the velocity-systems that Clifford has called right- and left-vectors (the Scime words tii'6 here applied to the finite displacements themselves). This deduction is materially aided by considering the system of equidistant surfaces of a given pair of conjugate lines, the two sets of generators on which constitute respectively the right-parallels and the left-parallels of the