Sotices respecting New Books. 



263 



to A, then the body will commence to twist about the screw 0, of 

 which r) is the polar with respect to the quadratic n-system 

 •composed of the imaginary screws about which the body would 

 twist with zero kinetic energy." 



Chapter xix. is taken up with explaining Homographic Screw 

 Systems, the importance of which will at once be recognized 

 by those who have read the iuimitable ' Dynamical Parable,' re- 

 printed in the Appendix. The chapter following contains an 

 account of Emanants and Pitch Invariants deduced analytically by 

 transformation of screw-coordinates ; and in Developments of the 

 Dynamical Theory (Chap, xx.) we are introduced to Chiastic 

 Homography * — a very fundamental conception. We simply 

 quote from the Geometrical Theory (Chap, xxi.) two enunciations 

 which speak for themselves : — "A body at rest, having n degrees 

 of freedom, is struck by an impulsive wrench upon a given screw. 

 It is required to construct the instantaneous screw about which 

 the body will commence to twist ; " and, " Given two systems of 

 screws of the third order. It is generally possible, in one way, 

 but only in one, to design, and place in a particular position a 

 rigid body such that, if that body, while at rest and unconstrained, 

 receive an impulsive wrench about any screw of the first system, 

 the instantaneous movement will be a twist about a screw of the 

 second system." 



Chapter xxin. is a collection of various exercises ; and the 

 next chapter contains the theory of Screw-chains. A mass-chain 

 consists of a number of rigid bodies fettered by constraints of the 

 most general type, and taken in an arbitrary but definite order. 

 In any possible small motion of the system let the twists of the 

 various bodies be observed. If we conceive the twists which the 

 first and second bodies receive to be compounded, we obtain what 

 may be called an intermediate twist. In like manner intermediate 

 twists may be found for every adjacent pair of bodies in the mass- 

 chain. The screws upon which the actual twists of the bodies 

 take place, taken in order and interpolated by the proper inter- 

 mediate screws upon which the intermediate twists may be 

 conceived to occur, constitute a screw-chain. Given three screws 

 on a cylindroid, the triangle of twists determines the ratios of the 

 amplitudes of the twists on three screws when one is the resultant 

 of the other two. Hence a determinate twist applied on the 

 first screw of a given screw-chain will be accompanied by deter- 

 minate twists on every member of the chain, and a determinate 

 motion of the mass-chain will correspond. Starting from these 

 conceptions, the author shows that a mass-chain capable of 

 twisting about two or more screw-chains can likewise be twisted 

 about every screw-chain linearly compounded from the given 

 chains. He extends the theory of homography to screw-chains, 



* In the case of Chiastic homography, if A and B be two screws, and 

 A', B' their correspondents ; then when A is reciprocal to B', B must be 

 reciprocal to A f . 



