229, 230] 



FREEDOM OF THE FIFTH ORDER. 



247 



To express this equation concisely we introduce two classes of subsidiary 

 magnitudes. We write one magnitude of each class as a determinant. 



= P. 



/82 



73 

 74 



75 



By cyclical interchange the two analogous functions Q and R are denned. 



- y 3 Cl 3&amp;gt; 



73 



74 



By cyclical interchange the two analogous functions M and JV^ are denned. 

 The equation for p reduces to 



The reduction of this equation to the first degree is an independent 

 proof of the principle, that one screw, and only one, can be determined 

 which is reciprocal to five given screws ; p being known, a, /3, 7 can be found, 

 and also two linear equations between x, y , z , whence the reciprocal screw is 

 completely determined. 



For the study of the screws representing a five-system we may take the 

 first screw of a set of canonical coreciprocals to be the screw reciprocal to 

 the system. Then the co-ordinates of a screw in the system are 



0, 2 , 6 3 , ... 6 , 



while if X, p, v be the direction cosines of 6 and x, y, z a point thereon, and 

 p e the pitch we have ( 43) 



(p e + a) cos \ z cos p + y cos v = 0. 



We can obtain at once the relation between the direction and the pitch 

 of the screw belonging to the system and passing through a fixed point. 

 If p e = and 2 and y be given, then the equation shows that the screw is 

 limited to a plane ( 110). 



230. Six Screws Reciprocal to One Screw. 



When six screws, A l} ... A 6 are reciprocal to a single screw T, a certain 



