Notices respecting New Books. 261 



reference to the Preface, or to the excellent and most useful series 

 of Bibliographical Notes (pp. 510-539) printed in the Appendix 

 to the new volume. To mention only two of the more important 

 additions : the theory of Screw-chains was not communicated toj 

 the Royal Irish Academy till 1881, nor the theory of Permanent 

 Screws till 1890, — the latter so fundamental in the present theory ;. 

 and the former so far-reaching in its applications, for "all the 

 instantaneous motions of every molecule in the universe are only a 

 twist about one screw-chain, while all the forces of the universe 

 are but a wrench upon another" (' A Dynamical Parable,' p. 509). 



After an introduction in which the displacement of a rigid 

 system is regarded as a particular nomographic transformation of 

 its points, the author proceeds in the first chapter to explain his 

 terminology, and to point out the restrictions he imposes on the 

 generality of his investigations. The second chapter deals with 

 the relations of a pair of screws, the composition of pairs of 

 twists and wrenches, the locus of axes of screws compounded from 

 a given pair, and the distribution of pitch. The equation of the 

 cylindroid is obtained, and cases of its degradation are pointed 

 out, fuller treatment being resumed in Chapters v. and xiii. We 

 notice (p. 510) that Hamilton must now be regarded as the 

 inventor of that celebrated surface, for his law of the Virtual Foci 

 of a congruency, published in 1830, implies its properties. In a 

 word, every ray of a congruency of the most general type adjacent 

 to a given ray intersects at right angles a generator of a certain 

 cylindroid, and the small angle it makes with the axis is equal to 

 the intercept on the generator divided by the pitch appropriate 

 thereto. This relation appears to us to establish in the most 

 conclusive manner the fundamental importance of the cylindroid 

 in all problems connected with systems of right lines. Yet if 

 Hamilton may justly be called the inventor of that surface, with 

 no less justice may we speak of Sir Robert Ball as its discoverer. 



The short third chapter is devoted to Reciprocal Screws * ; and 

 the fourth contains an account of Screw-coordinates and Co- 

 reciprocal Screws. We would willingly linger over this subject, 

 fascinated by the compactness and completeness of a system of 

 six co-reciprocals, about which we believe the last word has not 

 yet been said, and by the symmetry and adaptability of screw- 

 coordinates. For our own sake we regret the author has not 

 informed us of the part played by representative points in a space 

 of five or six dimensions, and has not illustrated for us the 

 application of screw-coordinates to questions usually treated by 

 Cayley's six coordinates of a line (p. 514). We miss, too, the 

 vivid light screw-conceptions are capable of shedding on the 

 complex. For instance, any point on a body free to twist about 

 a given screw describes a helix, and the tangent of the inclination 

 is equal to the pitch divided by the perpendicular on the axis. 



* A wrench does not disturb the equilibrium of a body only free to 

 twist about screws reciprocal to that on which the wrench is situated. 



