A Treatise on the Theory of Screws

522 THE THEORY OF SCREWS. more generally of two Dynames in Plücker’s sense or of two “motors” as Clifford prefers to call them. A “ motor” may be said to bear the same relation to a screw which a vector bears to a ray. The calculus of Biquaternions is generalized from that of quaternions and belongs to the non-Euclidian geometry. See Klein’s celebrated paper, “Ueber die sogenannte nicht Euclidische Geometrie.” Math. Ann., Band IV., pp. 573-625. This paper of Clifford’s has been the commence- ment of an extensive theory at which many mathematicians have since worked. Chap xxvi. discusses some of Clifford’s theorems and in the course of these bibliographical notes there are several references to this theory. See under the names of Everett, Padeletti, Cox, Heath, Buchheim, Cayley, Burnside, Joly, Kotelnikof, and M’Aulay. Ball (R. S.)—Researches in the Dynamics of a Rigid Body by the aid of the Theory of Screws. Second Memoir (June 19, 1873). Philosophical Transactions, pp. 15-40 (1874). The chief advance in this paper is expressed by the theorem that a rigid body lias just so many principal screws of inertia as it has degrees of freedom. This theorem is a generalization for all cases of a rigid system, no matter what be the nature and number of its constraints, of the well-known property of the principal axes of a rigid body rotating around a fixed point. It is shown that if the screws on one cylindroid be regarded as impulsive screws, the system of corresponding instantaneous screws lie on another cylindroid. Any four screws on the one cylindroid, and their four correspondents on the others are equi-anhannonic. This theorem leads to many points of connexion between theoretical dynamics and modern geometry. It has been greatly developed sub- sequently. A postscript to this paper gives a brief historical sketch which shows the rela- tion of the theory of screws to the researches of Pliicker and Klein on the Theory of the Linear Complex. Skatow.—Zusammenstellung der Sätze von den übrigbleibenden Bewegungen eines Körpers, der in einigen Punkten seiner Oberfläche durch normale Stützen unterstützt wird. Schlömilch’s Zeitschrift für Mathern, u. Physik, B. xviii., p. 224, 1873. Halphen—Sur le déplacement d’une solide invariable. Bulletin de la Soc. Math., Vol. ii., pp. 56-62 (23 July, 1873). The study of the displacements of a rigid body is distributed into six cases according to the number of degrees of freedom. This paper like so many others on the present subject has been suggested by the writings of M. Mannheim. It gives for instance a proof of Mannheim’s theorem that all the displacements of a solid restrained by four conditions could be produced by two rotations around two determinate lines. These are of course in our language the two screws of zero pitch on the cylindroid expi-essing the freedom. Halphen considers in some cases conditions more general than those of Mannheim and adds some theorems of quite a new class. Thus still referring to the case of a body restrained by four con- ditions, i. e. with two degrees of freedom, he shows how the movements of every point are limited to a surface, and then calling the two screws of zero pitch the “ axes” we have as follows. “ Les projections, sur un plan donné, des éléments superficiels, décrit par les points du corps, sont proportionelles aux produits des segments interceptés, sur des sécantes paralleles issues de ces points, par un para- boloide passant par les deux axes, et ay ant le plan donné pour plan directeur.”