368 THE THEORY OF SCREWS. [338- 



imposed by the constraints. The displacement of each element could, 

 however, have been effected by a twist of appropriate amplitude about a 

 screw specially correlated to that element. The total effect of the displace 

 ment could, therefore, have been produced by giving each element a certain 

 twist about a certain screw. 



339. The Graphic and Metric Elements. 



In the lowest type of freedom which the mass-chain can possess (short 

 of absolute fixity) the freedom is of the first order, and any position of 

 the mass-chain admits of specification by a single co-ordinate. In such a 

 case the screw appropriate to each element is unique, and is completely 

 determined by the constraints both in position and in pitch. The ratio 

 of the amplitude of each twist to the amplitudes of all the other twists is 

 also prescribed by the constraints. The one co-ordinate which is arbitrary 

 may be conveniently taken to be the amplitude of the twist about the 

 first screw. To each value of this co-ordinate will correspond a possible 

 position of the mass-chain. As the ratios of the amplitudes are all known, 

 and as the first amplitude is given, then all the other amplitudes are known, 

 and consequently the position assumed by every element of the mass-chain 

 is known. 



The whole series of screws and the ratios of the amplitudes thus embody 

 a complete description of the particular route along which the mass-chain 

 admits of displacement. The actual position of the mass-chain is found 

 by adding to the purely graphic element which describes the route a metric 

 element, to indicate the amplitude through which the mass-chain has 

 travelled along that route. This amplitude is the arbitrary co-ordinate. 



340. The Intermediate Screw. 



It will greatly facilitate our further progress to introduce a conventional 

 process, which will clearly exhibit the determinate character of the ratios 

 of the amplitudes in the screw series. Consider the two first screws, i and or 2 

 of the series. Draw the cylindroid (o. l , 2 ) which contains these two screws. 

 Since a l and 2 are appropriated to two different elements of the mass-chain, 

 no kinematical significance can be attached to the composition of the two 

 twists on j and 2 . If, however, the two twists on j and a 2 , having the 

 proper ratio of amplitudes, had been applied to a single rigid body, the dis 

 placement produced is one which could have been effected by a single 

 twist about a single screw on the cylindroid (a l} et 2 ). If this inter 

 mediate screw be given, the ratio of the amplitudes of the twists on 

 the given screws is determined. It is in fact equal to the ratio of the 

 sines of the angles into which the intermediate screw divides the angle 

 between the two given screws. With a similar significance we may conceive an 

 intermediate screw inserted between every consecutive pair of the p original 

 screws. 



