A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 531 The points which represent a series of impulsive screws and the points which represent the series of corresponding instantaneous screws are homographic. The three double points of the homography represent the three principal screws of inertia. The three harmonic screws about any one of which the body would oscillate for ever in the vicinity of a position, of stable equilibrium are determined as the vertices of the common conjugate triangle of two conics. This memoir is the basis of Chap. xv. in the present volume. Hyde (E. W.)—Annals oj Mathematics, Vol. iv., No. 5, p. 137 (1888). The author writes: “I shall define a screw to be the sum of a point-vector and a plane-vector perpendicular to it, the former being a directed and posited line, the latter the product of two vectors, hence a directed but not posited plane.” Prof. Hyde proves by his calculus many of the fundamental theorems in the present theory in a very concise manner. Gravelius (Harry)—Theoretische Mechanik starrer Systeme. Auf Grund der Methoden und Arbeiten und mit einem Vorworte von Sir Robert S. Ball. Berlin, 1889. 8vo., p. 619. The purport of this volume is expressed in the first paragraph of the preface : “Das vorliegende Werk stellt sich die Aufgabe, zusammenhängend und als Lehr- buch die in zahlreichen Arbeiten von Sir Robert Ball geschaffene Theorie der Mechanik starrer Systeme darzustellen. Es umfasst somit dem Inhalte nach sämmtliche Abhandlungen des Herrn Ball.” Thus the work is mainly a trans- lation of the Theory of Screws and of the subsequent memoirs up to the date 1889. Herr Gravelius has however added much, and his original contributions to the theory are specially found in Chap. xix. “ Projective Beziehungen räumlicher Schraubengebilde.” I feel very grateful to Herr Gravelius for his labour in render- ing an account of the subject into the German language. Zanchevsky (I.) — Theory of Screws and its Application to Mechanics, pp. i—xx., 1—131. Odessa, 1889. I must first acknowledge the kindness with which my friend Mr G. Chawner, Fellow of King’s College, has assisted me by translating the Russian in which this book is written. I here give some passages from the introduction. Zanchevsky remarks that in the Theory of Screws I omitted to give a proof of the reality of all the roots of the equation of the nth degree which determines the principal Screws of Inertia, and then he gives a proof derived from a theorem of Kronecker. “Zur Theorie der linearen und quadratischen Formen." Monats- berichte der Acad. der Wissenschaften zu Berlin, 1868, p. 339. The theorem is as follows. Let U and V be two homogeneous quadratic forms with n variables. If the. discriminant of \U + yV when equated to zero gives a single imaginary root then no member of the system \U + y.V can be expressed as the sum of n squares. We should, however, in this matter refer to the earlier paper of Weierstrass, p. 513. From this theorem Zanchevsky proves the reality of the roots of the Harmonic Determinant. (See § 85.) Then follows a discussion of the principal Screws of Inertia for a constrained system. Chap. I. contains an exposition of Plilcker’s theory of the linear complex of the 1st order. Here will be found the conception of the screw, its co-ordinates, the virtual coefficient of two screws, and the connection between the systems of vectors which determine reciprocal screws. He remarks that this connection may be directly derived from the works of Lornoff. 34—2