352] THE THEORY OF SCREW-CHAINS. 387 



t) by twists about which those displacements could have been effected. 



In the construction of an eighth chain, 0, we may proceed as follows : 



Choose any arbitrary screw lm Decompose a twist on 1 into components 

 on !, &, ry 1( $ lt GI , . This must be possible, because a twist about any 

 screw can be decomposed into twists about six arbitrary screws, for we 

 shall not discuss the special exception when the six screws belong to a system 

 of lower order. 



The twists on a^ ... , &c., determine the twists on the screw-chains 

 ,... and therefore the twists on the screws a 2 , ... ,, which compound 

 into a twist on &amp;lt;9 2 , similarly for 3 , &c.; consequently a screw-chain of which 

 #! is the first screw, and which belongs to the system, has been constructed. 

 This is, however, only one of a number of screw-chains belonging to the 

 system which have 6 l for their first screw. The twist on l might have been 

 decomposed on the six screws, &, 7l , S lt l , , 77,, and then the screws 2 , &c., 

 might have been found as before. These will of course not be identical with 

 the corresponding screws found previously. Or if we take the whole seven 

 screws, o^, ... 77!, we can decompose a twist on 1 in an infinite number of 

 ways on these seven screws. We may, in fact, choose the amplitude of the 

 twist on any one of the screws of reference, oq , for example, arbitrarily, and 

 then the amplitudes on all the rest will be determined. It thus appears 

 that where l is given, the screw 2 is not determined in the case of freedom 

 of the seventh order ; it is only indicated to be any screw whatever of a 

 singly infinite number. The locus of 2 is therefore a ruled surface ; so will 

 be the locus of 3 , &c. and we have, in the first place, to prove that all these 

 ruled surfaces are cylindroids. 



Take three twists on 1} such that the arithmetic sum of their amplitudes 

 is zero, and which consequently neutralize. Decompose the first of these 

 into twists on a 1} &, 7l , S l} , ^, the second on a,, ft, 7l , 8 1} e ,, 17,, and the 

 third on a 1} ft, %, 8 ly e 1} . It is still open to make another supposition 

 about the twists on 6^ let us suppose that they are such as to make the two 

 components on ^ vanish. It must then follow that the total twists on each 

 of the remaining six screws, viz. 1} ft, 7l) 8^ e ly shall vanish, for their 

 resultant cannot otherwise be zero. All the amplitudes of the twists about 

 the screw-chains of reference must vanish, and so must also the amplitudes 

 of the resultant twists when compounded. We should have three different 

 screws for 2 corresponding to the three different twists on 0^ and as the 

 twists on these screws must neutralize, the three screws must be co- 

 cylindroidal. 



We can, therefore, in constructing a screw-chain of this system, not only 

 choose #! arbitrarily, but we can then take for 6 2 any screw on a certain 

 cylindroid : this being done, the rest of the screw-chain is fixed, including 

 the intermediate screws. 



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