386 THE THEORY OF SCREWS. [351, 



351. Freedom of the sixth order. 



In freedom of the sixth order we select at random six displacements of 

 which the mass-chain admits, and then construct the six corresponding screw- 

 chains. The homologous screws in this case lie on six-systems, but a six- 

 system means of course every conceivable screw. It is easily shown ( 248) 

 that if to one screw in space corresponds another screw, and conversely, then 

 the homography is completely established when we are given seven screws 

 in one system, and the corresponding seven screws in the other. Any eighth 

 screw in the one system will then have its correspondent in the other imme 

 diately determined. 



It is of special importance in the present theory to dwell on the type of 

 homography with which we are here concerned. If on the one hand it 

 seems embarrassing, from the large number of screws concerned, on the 

 other hand we are to recollect that the question is free from the complication 

 of regarding the screws as residing on particular n-systems. Seven screws 

 may be drawn anywhere, and of any pitch ; seven other screws may also be 

 chosen anywhere, and of any pitch. If these two groups be made to corre 

 spond in pairs, then any other screw being given, its corresponding screw will 

 be completely determined. Nor is there in this correspondence any other 

 condition, save the simple one, that to one screw of one system one screw 

 of the other shall correspond linearly. 



Six screw-chains having been found, a seventh is to be constructed. 

 This being done, the construction of as many screw-chains as may be desired 

 is immediately feasible. From the homographic relations just referred to 

 we have appropriate to each element of the system seven homologous screws, 

 and also appropriate to each consecutive pair of elements we have the seven 

 homologous intermediate screws. An eighth screw, appropriate to any 

 element, may be drawn arbitrarily, and the corresponding screw being con 

 structed on each of the other systems gives at once another screw-chain about 

 which the system must be free to twist. 



When a mass-chain has freedom of the sixth order we see that any one 

 element may be twisted about any arbitrary chosen screw, but that the 

 screw about which every other element twists is then determined, and so 

 are also the ratios of the amplitudes of the twists, by the aid of the inter 

 mediate screws. 



352. Freedom of the seventh order. 



Passing from the case of six degrees of freedom to the case of seven 

 degrees, we have a somewhat remarkable departure from the phenomena 

 shown by the lower degrees of freedom. Give to the mass-chain any seven 

 arbitrary displacements, and construct the seven screw-chains, a, /3, 7, B, e, 



