392 THE THEORY OF SCREWS. [355, 



Of two reciprocal screw-chain systems, each expresses the collection of 

 ivrench-chains of which each one will equilibrate when applied to a mass-chain 

 only free to twist about all the chains of the other system. 



This is, perhaps, one of the most comprehensive theorems on Equilibrium 

 which could be enunciated. 



356. Impulsive screw-chains and instantaneous screw-chains. 



Up to the present we have been occupied with considerations involving 

 kinematics and statics : we now show how the principles of kinetics can be 

 illustrated by the theory sketched in this chapter. 



We shall suppose, as before, that the mechanical arrangement which we 

 call the mass-chain consists of jj, elements, and that those elements are so 

 connected together that the mass-chain has n degrees of freedom. We shall 

 also suppose that the mass-chain is acted upon by a wrench about any screw- 

 chain whatever. The first step to be taken is to show that the given 

 wrench-chain may be replaced by another more conveniently circumstanced. 

 Take any n chains of the given system, and 6/u, n chains of the reciprocal 

 system, then the given wrench-chain can be generally decomposed into 

 components on the n + (6/z - n) chains here specified. The latter, being all 

 capable of neutralization by the reaction of the constraints, may be omitted, 

 while the former n wrench-chains admit of being compounded into a single 

 wrench-chain. We hence have the following important proposition : 



Whatever be the forces which act on a mass-chain, their effect is in general 

 equivalent to that of a wrench on a screw-chain which belongs to the system of 

 screw-chains expressing the freedom of the mass-chain. 



The application of this theorem is found in the fact that, while we still 

 retain the most perfect generality, it is only necessary, either for twists or 

 wrenches, to consider the system, defined by n chains, about which the mass- 

 chain can be twisted. 



Let us consider the mass-chain at rest in a specified position, and suppose 

 it receives the impulsive action of any set of forces, it is required to determine 

 the instantaneous motion which the system will acquire. The first operation 

 is to combine all the forces into a wrench-chain, and then to transform that 

 wrench-chain, in the manner just explained, into an equivalent wrench- 

 chain on one of the screws of the system. Let 6 be the screw-chain of the 

 system so found. In consequence of this impulsive action the mass-chain, 

 previously supposed to be at rest, will commence to move ; that motion can, 

 however, be nothing else than an instantaneous twist velocity about a screw- 

 chain a. We thus have an impulsive screw-chain 6 corresponding to an 

 instantaneous screw-chain a. In the same way we shall have the impulsive 

 screw-chains &amp;lt;, i/r, &c., correlated to the instantaneous chains, /3, y, &c. 



