384 THE THEORY OF SCREWS. [349, 



on the cylindroid to which it is confined its specification merely gives a 

 single datum. To be given a pair of corresponding Dynamos is, however, 

 to be given really two data one of these is for the screws themselves as 

 before, while the other is derived from the ratio of the amplitudes. Thus 

 while three pairs of corresponding screws amount to three data, two pairs 

 of corresponding Bynames amount to no less than four data ; the additional 

 datum in this case enabling us to indicate the intensity of each correspondent 

 as well as the screw on which it is situated. 



It can further be shown in the most general case of the correspondence 

 of the Bynames in two w-systems that the number of pairs of Bynames 

 required to define the correspondence is one less than the number of pairs 

 of screws which would be required to define merely a screw correspondence 

 in the same two w-systems. In an n -system a screw has n l disposable 

 co-ordinates. To define the correspondence we require n + 1 pairs of screws. 

 Of course those on the first system may have been chosen arbitrarily, so 

 that the number of data required for the correspondence is 



A Byname in an w-system has n arbitrary data, viz., n 1 for the screw, 

 and one for the intensity : hence when we are given n pairs of corresponding 

 Bynames we have altogether n 2 data. We thus see that the n pairs of 

 corresponding Bynames really contribute one more datum to the problem 

 than do the n + l pairs of corresponding screws. The additional datum is 

 applied in allotting the appropriate intensity to the sought Byname. 



We can then use either the n pair of Byname correspondents or the 

 (n + l) pairs of screw correspondents. In previous articles we have used the 

 latter; we shall now use the former. 



350. Freedom of the fifth order. 



In the higher orders of freedom the screw correspondence does not indeed 

 afford quite so simple a means of constructing the several pairs of corre 

 sponding screws as we obtain by the Byname correspondence. In two 

 five-systems the correspondence is complete when we are given five Bynames 

 in one and the corresponding five Bynames in the other. To find the 

 Byname X in the second system, corresponding to any given Byname A 

 in the first system, we proceed as follows: Decompose A into Bynames 

 on the five screws which contain the five given Bynames on the first system. 

 This is always possible, and the solution is unique. These components will 

 correspond to determinate Bynames on the five corresponding screws : these 

 Bynames compounded together will give the required Byname X both in 

 intensity and position. 



In the general case where a mass-chain possesses freedom of the fifth 



