A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 523 Lindemann (F.)—Ueber unendlich kleine Bewegungen und Uber Kraftsysteme bei allgemeiner projectivischer Massbestimmung. Math. Ann., V ol. vii., pp. 56-143 (July, 1873). This is a memoir upon the statics and kinematics of a rigid body in elliptic or hyperbolic space. Among several results closely related to the Theory of Screws, we find that the cylindroid is only the degraded form in parabolic or common space of a surface of the fourth order, with two double lines. Lindemann both by this memoir and by that entitled “ Projectivische Behandlung der Mechanik starrer Körper” in the same volume has become the pioneer of an immense and most attractive field of exploration. He has laid down the principles of Dynamics in Non-Euclidian space. One small part of this subject I have endeavoured to develop in Chap. xxvi. Weiler (A.)—Ueber die verschiedenen Gattungen der Complexe zweiten Grades. Math. Ann., Vol. vii,, pp. 145—207 (July, 1873). In this elaborate memoir the author enumerates fifty-eight different species of linear complexes of the second order. The classification is based upon Kummer’s surface, which defines the singularities of the complex. These investigations are of importance in the present subject because, to take a single instance, the screws of a system of the fourth order form a linear complex of the second order. This complex is of a special type included among the 58 species. Ball (R. S.)—Screw Co-ordinates and their applications to problems in the Dynamics of a Rigid Body. Third memoir. Transactions of the Royal Irish Academy, Vol. xxv., pp. 259-327 (January 12, 1874). The progress of the present theory was much facilitated by the introduction of screw co-ordinates. The origin and the use of such co-ordinates are here explained. It is, however, to be understood that screw co-ordinates, though no doubt arrived at independently, ought properly to be regarded as an adaptation for dynamical purposes of Klein’s co-ordinates of a linear complex referred to six fundamental complexes, of which each pair are in involution or reciprocal, as we say in the terminolog}7 of this volume. The pitch of a screw a as expressed in terms of its six co-ordinates a1, ... a6 is where p ... p6, &c. are the pitches of the co-reciprocal screws of reference. The virtual coefficient of two screws a and ß is In the dynamical part of the subject the chief result of this paper is the fundamental theorem that, when the six screws of reference are the six principal screws of inertia, then p O1) paif ... PiPs are the co-ordinates of the impulsive wrench which will make the body commence to move by twisting about the screw a, ... a6. This was, perhaps, all that could be desired in the way of a simple connexion between an impulsive screw and the corresponding instantaneous screw, so far as their co-ordinates were concerned. Long before this paper was published I had been trying to find a geometrical connexion between two such screws which would exhibit their relation in a graphic manner. But the search was not to be successful until the results in the Twelfth Memoir were arrived at. Everett (J. D.)—On a new method in Statics and Kinematics. (Part I.) Messenger of Mathematics. New Series. No. 39 (1874), 45, 53 (1875). The papers contain applications of quaternions. The operator w + V<r ( ) is a tc motor,” us and <r being vectors, the former denoting a translation or couple, the