A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 521 cylindroid, or rather rediscoverers, for neither of them was-the earliest discoverer, for as shown in p. 510 the cylindroid was first introduced into science by Sir W. R. Hamilton so long ago as 1830. It is worthy of note that three investi- gators, and if I may add my own name a fourth also, following different lines of research, have each been independently led to perceive the importance of this particular surface in various theories of systems of lines. In the paper now before us I had developed the doctrine of reciprocal screws which is of such fundamental importance in the theory. I had arrived at this doctrine independently, and not until after the paper was printed did I learn that the essential conception of Reciprocal Screws had been announced by Professor Klein a few months before my paper was read (pp. 17, 18, 520). These facts have to be mentioned in explanation of the circumstance that this first paper contains no references to the names of either Plücker and Battaglini or Hamilton and Klein. Somoff (J.)—Sur les vitesses virtuelies d’une figure invariable, assujetties ä des equations de conditions quelconques de forme lineaire. St Pétersb. Acad. Sei. Bull., xviii., 1873, col. 162—184. This paper is an important one in the history of the subject. Its scope may be realized from the paragraph here quoted. “ Dans le mémoire que j’ai 1’honneur de presenter a 1’Académie je donne un moyen analytique pour determiner les vitesses virtuelles d’une figure invariable, en supposant que ces vitesses doivent satisfaire ä des equations de condition de la forme générale que je vients de citer. Je prends en meme temps en considération les propriétés des complexes Hnéaires de Plücker, auxquel les vitesses virtuelles d’une figure invariable sont intimement liées.” The analytical development of the Theory of the Constraints which follows is founded upon the conventions proposed by M. Résal in his “ Traité de Cinématique pure.” M. Somoff studies conditions of constraint which he has generalized from M. Mannheim’s “Étude sur le deplacement d’unefigure deforme invariable” (p. 519). It is instructive to read M. SomofFs paper in the light of the Theory of Screws. For example on p. 179 lie gives the theorem that every system of “ virtual velocities” which satisfies three linear equations can be produced by two rotations around two rays common to the three corresponding linear complexes. In our language we express this by saying that any displacement of a body with three degrees of freedom can be produced by rotation around two screws of zero pitch belonging to the system. This is easily seen, for let 0 be the screw about which the required displacement is a twist. Let </> be any other screw of the three- system, then the two screws of zero pitch on the cylindroid (0, </>) are two axes of rotation that fulfil the required condition. The cases of four and five degrees of freedom are also briefly discussed by Somoff, but without the conception of screw motion which he does not employ the results are somewhat complicated. Reference may also be made to Somoff, “ Theoretische Mechanik," translated from the Russian by A. Ziwet, Leipzig, 1878—9. Clifford (W. K.)—Preliminary Sketch of Biquaternions. Proceedings of the London Mathematical Society, .Nos. 64, 65, Vol. iv., pp. 381—395 (12th June, 1873). This is one of the modern developments of that remarkable branch of mathe- matics with which the names of Lobachevsky and Bolyai are specially associated. A Biquaternion is defined to be the ratio of two twists or two wrenches or