CHAPTER XXIV. 



THE THEORY OF SCREW-CHAINS*. 



338. Introduction. 



In the previous investigations of this volume the Theory of Screws has 

 been applied to certain problems in the dynamics of one rigid body. I 

 propose to show in the present chapter to what extent the conceptions 

 and methods of the Theory of Screws may be employed to elucidate certain 

 problems in the dynamics of any material system whatever. 



By such a system I mean an arbitrary arrangement of /j, rigid bodies 

 of any form or construction, each body being either entirely free or con 

 strained in any manner by relations to fixed obstacles or by connexions 

 of any kind with one or more of the remaining p 1 pieces. 



For convenience we may refer to the various bodies in the system by 

 the respective numerals 1, 2, ... /n. This numbering may be quite arbitrary, 

 and need imply no reference whatever to the mechanical connexions of 

 the pieces. The entire set of material parts I call for brevity a mass-chain, 

 and the number of the bodies in a mass-chain may be anything from unity 

 to infinity. 



I write, as before, of only small movements, but even with this limitation 

 problems of equilibrium, of small oscillations and of impulsive movements 

 are included. By the order of the freedom of the mass-chain, I mean the 

 number of generalized co-ordinates which would be required to specify a 

 position which that mass-chain was capable of assuming. The order cannot 

 be less than one (if the mass- chain be not absolutely fixed), while if each 

 element of the mass-chain be absolutely free, the order will be as much 

 as 6/i. 



Starting from any arbitrary position of the mass-chain, let it receive a 

 small displacement. Each element will be displaced from its original 

 position to an adjacent position, compatible of course with the conditions 



* Transactions Royal Irish Acad. Vol. xxvm. p. 99 (1881). 



