A Treatise on the Theory of Screws

526 THE THEORY OF SCREWS. Constraint of the most general nature cannot, however, be so produced. It is sufficient to mention that in the case of freedom of the fifth order the screw reciprocal to the system must have zero pitch, if the constraint is of the nature supposed by Schell, while in the general case the pitch may have any value. Ball (R. S.)—On Homographic Screw Systems. Proceedings of the Royal Irish Academy, Ser. 2, Vol. iii. p. 435 (1881). The theory of Homographic Screws shows the connection between certain geometrical theories of an abstract nature and Dynamics. The intimate alliance . between geometry and the higher branches of Rigid Dynamics is illustrated in this paper. Invariant functions of eight screws are studied, and a generalized type of homographic ratio involving eight screws is considered. (See Chap, xix.) Ball (R. S.)—On the Elucidation of a question in Kinematics by the aid of Non- Euclidian Space. Report of British Association, York, 1881, p. 535. Certain peculiarities which presented themselves in the geometrical representa- tion of the screws of a three-system by points in a plane are here shown to be due to the conventions of Euclidian space. The screws of a three-system in non- Euclidian space can be arranged in equal pitch hyperboloids, which have eight common points and eight common tangent planes. In Euclidian space the cor- responding quadrics are inscribed in a common tetrahedron and pass through four common points as explained in Chap. xv. Ball (K. S.) Certain Problems in the Dynamics of a Rigid System moving in Elliptic Space. (Fifth Memoir.) Transactions of the Royal Irish Academy Vol. xxviii., pp. 159-184 (1881). Ilie chief theorem proved in this paper is, that though the virtual moment of two homonymous vectors is zero only when the two vectors are “ rectangular,” yet the virtual moment of two heteronymous vectors is always zero. I may here mention another memoir which bears on the same subject. The title is, On the Theory of the Content.—Transactions of the Royal Irish Academy Vol. xxix., pp. 123-181 (1887). In this it is shown that the order in which two heteronymous vectors in elliptic space are applied to a rigid system may be inverted without affecting the result, which is, however, not a vector at all. On the other hand, when two homonymous vectors in elliptic space are applied to a rigid system, the result is, in every case, a homonymous vector ; but then the order of application could not be inverted without changing the result. These papers have contributed to Chap. xxvi. of the present volume. Padeletti (Dino)—Osservazioni sulla teoria delle dinami (Theory of Screws). Rencliconto della R. Accademia di Scienze Fis. e Nat. di Napoli Fascicolo 2° Feb. 1882. The author here gives a general account of the Theory of Screws so far as it had been developed up to 1876. The method he has employed for deducing the equations of the cylindroid is novel and instructive. The same author in the same journal for May 1882 has a paper entitled, Su un Calcolo nella teoria delle dinami analogo a quello del quaternioni.