342] THE THEORY OF SCREW-CHAINS. 369 



341. The definition of a Screw-chain. 



It will be convenient to have a name which shall concisely express the 

 entire series of /A original screws with the //, 1 intermediate screws whose 

 function in determining the amplitudes has just been explained. We may 

 call it a screw-chain. A twist about a screw-chain will denote a displace 

 ment of a mass-chain, produced by twisting each element about the 

 corresponding screw, through an amplitude whose ratio to the amplitudes 

 on the two adjacent screws is indicated by the intermediate screws. The 

 amplitude of the entire twist will, as already mentioned, be most conveniently 

 expressed by the twist about the first screw of the chain. We hence have 

 the following statement : 



The most general displacement of which a mass-chain is capable can be 

 produced by a twist about a screw-chain. 



342. Freedom of the first order. 



Given a material system of p elements more or less connected inter se, 

 or related to fixed points or supports : let it be required to ascertain the 

 freedom which this material system or mass-chain enjoys. The freedom is to be 

 tested by the capacity for displacement which the mass-chain possesses. As 

 each such displacement is a twist about a screw-chain, a complete examina 

 tion of the freedom of the mass-chain will require that a trial be made to 

 twist the mass-chain about every screw-chain in space which contains the 

 right number of elements /u,. If in the course of these trials it be found that 

 the mass-chain cannot be twisted about any screw-chain, then the system 

 is absolutely rigid, and has no freedom whatever. If after all trials have 

 been made, one screw-chain, and only one, has been discovered, then the 

 mass-chain has freedom of the first order, and we have the result thus 

 stated : 



When a mass-chain is so limited by constraints, that its position can be 

 expressed by a single co-ordinate, then the mass-chain is said to have freedom 

 of the first order, and its possible movements are solely those which could be 

 accomplished by twisting about one definite screw-chain. 



By this method of viewing the question we secure the advantage of 

 eliminating, as it were, the special characters of the constraints. The 

 essential moving parts of a steam-engine, for example, have but one degree 

 of freedom. Each angular position of the fly-wheel necessarily involves a 

 definite position of all the other parts. A small angular motion of the fly 

 wheel necessarily involves a definite small displacement of each of the 

 other parts. Complicated as the mechanism may be, it is yet always possible 

 to construct a screw-chain, a twist about which would carry each element 

 from its original position into the position it assumes after the displacement 

 has been effected. 



B. 24 



