A Treatise on the Theory of Screws

530 THE THEORY OF SCREWS. two principal screws of inertia. This paper is the development of an earlier one. See Proceedings of the Royal Irish Academy, 2nd Series, Vol, iv. p. 29. The substance of it has been reproduced in Chaps v. and xn. of the present volume. Ball (R. S.)—On the Plane Sections of the Cylindroid. Seventh Memoir. Trans, of the Royal Irish Academy, Vol. xxix., pp. 1-32 (1887). This is a geometrical study of the cylindroid regarded as a conoidal cubic with one nodal line and three right lines in the plane at infinity. Plane sections of the cylindroid are shown in plates drawn to illustrate calculated cases. It is shown that the chord joining the points in which two reciprocal screws intersect a fixed plane envelops a hyperbola which has triple contact with the cubic curve in which the fixed plane cuts the cylindroid. See Chap. xni. of the present volume. I may take this opportunity of mentioning in addition to what has been said on the subject of models of the cylindroid in Chap. xm. that very simple and effective models of this surface can now be obtained from Martin Schilling. Halle a Saale. See his catalogue for Feb. 1900. Roberts (R. A.)—Educational Times, xlvi. 32-33 (1S87). In this it is shown that under the circumstances described the shadow of the cylindroid z (x2 + y2) — 2mxy = 0 on the plane z = 0 exhibits the hypocycloid with three cusps. Ball (R. S.)—A Dynamical Parable, being an Address to the Mathematical and Physical Section of the British Association. Manchester, 1887. This has been given in Appendix n. p. 496. It may be added here that it has been translated into Hungarian by Dr A. Seydler, and into Italian by G. Vivanti. Tarleton (F. A.)—On a new method of obtaining the conditions fulfilled when the Harmonic Determinant has equal roots. Proceedings of the Royal Irish Academy, 3rd Series, Vol. i. No. 1, p. 10 (1887). This discusses the case of equal roots in the harmonic determinant so important in the Theory of Screws as in other parts of Dynamics. It should be studied in connection with § 85 of the present volume; also Note n. p. 484. See also Zanchevsky, p. 531. Ball (R. S.)—How Plane Geometry illustrates general problems in the Dynamics oj a Rigid Body with Three degrees of Freedom. Eighth Memoir. Transactions of the Royal Irish Academy, Vol. xxix., pp. 247-284 (1888). The system of the third order is of such special interest that it is desirable to have a concise method of representing the screws which constitute it. We here show that the screws of such a system correspond to the points in a plane. This is the development of an earlier paper communicated to the Royal Irish Academy in 1881. Proceedings, 2nd Series, Vol. iii., pp. 428-434. In this method of representation the screws on a cylindroid belonging to the system are represented by the points on a straight line. The screws of any given pitch will have as their correspondents the points on a certain conic. A pair of points conjugate to the conic of zero pitch will correspond to a pair of reciprocal screws. The conic which represents the screws of zero pitch, and the conic which represents the screws of infinite pitch, will have a common conjugate triangle. The vertices of that triangle correspond to the principal screws of the system. It is proved that the pitch quadrics of a three-system are all inscribed in a common tetrahedron and have four common points on the plane at infinity.