A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 529 Minchin (G. M.)—Treatise on Statics. 3rd edition, Vol. ii. (1886). In pp. 17—43 of this standard work the Theory of Screws is discussed. An instructive construction for the cylindroid is given on p. 20. We may also note the following theorem proved on p. 25, “If the wrench on any screw of the cylindroid is replaced by a force and a couple at the centre of the pitch-conic (centre of the cylindroid) the axis of this couple will lie along the perpendicular to the diameter of the pitch-conic which is conjugate to the direction of the force— or in other words, the plane of the couple will be that of the axis of the cylindroid and this conjugate diameter.” Schönflies (Arthur)—Geometrie der Bewegung in synthetischer Darstellung, pp. 1—194, 8vo. Leipzig, 1886. The third chapter of this work, pp. 79-192, is devoted to the geometrical study of the movement of a rigid system. The author uses the word parameter to express what we have designated as the pitch. As an illustration of the theorems given I cite the following from p. 92, “ Bewegt sich ein unveränderliches System beliebig im Raume, und ist in irgend einem Augenblick eine Gerade desselben senkrecht zur Tangente der Bahn eines ih/rer Punkte, so ist sie es zu den Bahntangenten aller Punkte In the language of the present volume in which the dynamical and kinetical conceptions are so closely interwoven, this theorem appeal's as follows. Le^ two screws a and ß be reciprocal and let the pitch of a be zero. A twist of a rigid body about ß can do no work against a force on a. But a may be considered to act on the rigid body at any point in its line of application. Hence the displace- ments of every such point must be perpendicular to a. The following suggestive theorems may be quoted from pp. 116, 117 : “Die sämmtlichen Punkte des Systems deren Bahnen nach einem festen Punkte D des Raumes gerichtet sind, liegen in jedem Augenblick auf einer Raum- curve dritter Ordnung 0.” “Die Raumcurve C enthält die unendlich fernen imaginären Kreispunkte der zur Axe der Schi-aubenbewegung senkrechten Ebenen.” This work contains indeed much that it would be interesting to quote. I must however content myself with one more remark from p. 153, which I shall give in our own terminology. When a rigid body has freedom of the second order it can of course be twisted about any screw on a cylindroid. Such a twist can always be decomposed into two rotations around the two screws of zero pitch P and Q. The rotation around P does not alter P. Hence whatever be the small displace- ment of the system the movement of P can never be other than a rotation around Q, and the movement of Q can never be other than a rotation around P. Ball (R. S.)—Dynamics and Modem Geometry: a new chapter in the Theory of Screws. Sixth Memoir. Cunningham Memoirs of the Royal Irish Academy, No. iv., pp. 1-44 (1886). We represent the several screws on the cylindroid by points on the circum- ference of a circle. The angle between two screws is the angle which their corresponding points subtend at the circumference. The shortest distance of any two screws is the projection of the corresponding chord on a fixed ray in the plane of the circle. Any chord passing through the pole of this ray intersects the. circle in points corresponding to reciprocal screws. The pitch of any screw is the distance of its cori’esponding point from this ray. A system of points representing instantaneous screws and the corresponding system representing the impulsive screws are homographic. The double points of the homography correspond to the