262 .Notices respecting New Books. 



Any line through the point at right angles to the direction of 

 motion belongs to a linear complex, and in particular the line 

 touching the cylinder containing the helix of motion and cutting 

 that helix at right angles. The rays of the complex consequently 

 envelope helices orthogonal to the helices of motion, and the 

 tangent of the angle of inclination of an enveloped helix is the 

 radius of the cylinder divided by the pitch. However, it is easy 

 to suggest omissions : and the present reviewer willingly admits 

 that an author who has made his subject pre-eminently his own is 

 better fitted than any reviewer, however self-confident he may be, 

 to select those portions most worthy of his treatment. The fact 

 is, the theory of screws is a large subject, and a book as large as 

 this is not sufficient to contain it all. 



The sixth chapter treats of the equilibrium of a rigid body, of 

 reciprocal systems, and of the number of parameters they involve. 

 Chapters vn., viii., ix., and xxv. may be conveniently grouped 

 together, as they deal with Principal Screws of Inertia, Screws of 

 the Potential, Harmonic Screws, and Permanent Screws respec- 

 tively. There is a reason for the wide gap between Chapters vn. 

 and xxv., as the latter chapter is treated by the method of screw- 

 chains, but the subjects are nevertheless very closely related. 

 When a rigid body has partial freedom it is possible to find a 

 limited number of screws so that a wrench administered on any 

 one of these will cause the body to begin to twist about that 

 screw ; a limited number of screws may also be found enjoying the 

 property that if the body is set twisting about any one it will 

 continue for a moment twisting about the same screw. These are 

 Principal Screws of Inertia and Permanent Screws respectively, 

 and though generally quite distinct they coalesce in the case of a 

 body having one point fixed (p. 400). We draw attention to this 

 subject as it is of the greatest interest and importance, and we 

 believe further developments may be made, for example in the 

 case of indeterminate screws, for JSTote II. (p. 484) does not quite 

 satisfy us. When a body is twisted from a position of stable 

 equilibrium, the restoring wrench will lie upon the screw of the 

 twist if it is a Principal Screw of the Potential (Chap, viii.) ; and 

 if the body is allowed to oscillate it will perform simple harmonic 

 twists upon a Harmonic Screw alone (Chap. ix.). 



The cases of freedom of various orders occupy separate Chapters 

 (x. to xviii.), plane representations of freedom of the second and 

 third order being specially treated in Chapters xn. and xv. in very 

 considerable detail. We are struck by the care with which the 

 figure on p. 211 is drawn to scale, as, indeed, is the case with 

 several other diagrams representing a plane section of the Cylin- 

 droid, &c. In connection w T ith freedom of the fourth order, Sir 

 Kobert Ball touches the fringe of what seems to be an extensive 

 subject, and gives remarkable dynamical applications of the polar 

 screws of a quadratic n-system : — " A quiescent rigid body is free 

 to twist about all the screws of an enclosing (n-f-l)-system A. 

 If the body receive an impulsive wrench on a screw >? belonging 



