236] 



FREEDOM OF THE FIFTH ORDER. 



255 



Then if 6 be a screw of the five-system with direction cosines cos X, 

 cos //,, cos v, and if as, y, z be a point on the screw 6 and p e its pitch we must 

 have ( 216) 



( p e + a) cos X + z cos /* y cos v = 0. 



Fig. 41. 



Everything that we wish to specify about the five-system may be 

 conveniently inferred from this equation. 



For example, let it be desired to find the locus of the screws of a five- 

 system which can be drawn through a given point as , y , z and have the 

 given pitch p e . 



We have 



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



If os, y, z be a point on 6 we may substitute x x, y y, z z for cos X, 

 cos ft, cos v, and we obtain 



whence we see that the locus is a plane, as has been already proved other 

 wise. (When the pitch is zero, this is Mb bius theorem, 110.) 



If we change the origin to some other point P which may with complete 

 generality be that point whose co-ordinates are o, h, o and call X, Y, Z the 

 co-ordinates with these new axes, the equation becomes 



(p e + a)c 



