378 THE THEORY OF SCREWS. [346- 



In the choice of a screw-chain about which a mass-chain with four 

 degrees of freedom can twist there are three arbitrary elements. We may 

 choose as the first screw of the chain any screw from a given four-system. 



If one screw of the chain be moved over a two-system, or a three-system 

 included in the given four-system, then every other screw of the chain will 

 also describe a corresponding two-system or three-system. 



347. Freedom of the fifth order. 



In discussing the movements of a system which has freedom of the fifth 

 order, the analogies which have hitherto guided us appear to fail. Homo- 

 graphic pencils, planes, and spaces have exhibited graphically the relations 

 of the lower degrees of freedom ; but for freedom of the fifth degree these 

 illustrations are inadequate. No real difficulty can, however, attend the 

 extension of the principles we have been considering to the freedom of the 

 fifth order. We can conceive that two five-systems are homographically 

 related, such that to each screw on the one corresponds one screw on the 

 other, and conversely. To establish the homography of the two systems it 

 will be necessary to know the six screws on one system which correspond to 

 six given screws on the other : the screw in either system corresponding to 

 any seventh screw in the other is then completely determined. 



In place of the methods peculiar to the lower degrees of freedom, we 

 shall here state the general analytical process which is of course available in 

 the lower degrees of freedom as well. 



A screw 6 in a five-system is to be specified by five co-ordinates 1 , # 2 , B 3 , 

 64, B 5 . These co-ordinates are homogeneous; but their ratios only are con 

 cerned, so they are equivalent to four data. The five screws of reference 

 may be any five screws of the system. Let &amp;lt;f&amp;gt; be the screw of the second 

 system which is to correspond to 6 in the first system. The co-ordinates of 

 &amp;lt;f&amp;gt; may be referred to any five screws chosen in the second system. It will 

 thus be seen that the five screws of reference for &amp;lt;j&amp;gt; are quite different from 

 those of 6. 



The geometrical conditions expressing the connection between &amp;lt;/&amp;gt; and 9 

 will give certain equations of the type 



where t/j, ..., U 5 are homogeneous functions of B lt ..., 9 5 . These equations 

 express that one 9 determines one &amp;lt;f&amp;gt;. As however one &amp;lt; is to determine one 

 9 we must have also equations of the type 



where //, ..., U 6 are functions of &amp;lt;f&amp;gt; 1} ..., &amp;lt;/&amp;gt; 5 . 



From the nature of the problem these functions are algebraical and as 

 they must be one valued they must be rational functions. We have therefore 



