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 conies. 



This memoir is the basis of Chap. xv. in the present volume. 



HYDE (E. W.) Annals of Mathematics, Vol. iv., No. 5, p. 137 (1888). 



The author writes: &quot;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.&quot; 

 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 : 

 &quot;Das vorliegende Werk stellt sich die Aufgabe, zusammenhangend 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 

 sammtliche Abhandlungen des Herrn Ball.&quot; 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. &quot; Projective Beziehungen raumlicher 

 Schraubengebilde.&quot; 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 Screivs and its Application to Mechanics, pp. I xx., 

 1131. 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 whicli 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 wth degree which determines the 

 principal Screws of Inertia, and then he gives a proof derived from a theorem 

 of Kronecker. &quot; Zur Theorie der linearen und quadratischen Formen.&quot; Monats- 

 berichte der Acad. der Wissenschaften zu Berlin, 1863, p. 339. The theorem is as 

 follows. Let U and V be two homogeneous quadratic forms with u variables. If 

 the discriminant of \U + p,V when equated to zero gives a single imaginary root 

 then no member of the system \U + p.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 Plucker 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 Lorrioff. 



342 



