225] FREEDOM OF THE FOURTH ORDER. 237 



the screw 6 are subjected to no other relation than that implied by the 



quadratic relation 



U e = Q. 



As before we may substitute for lt ..., 6 6 from the equations 

 2a#j = (a 4- p e ) cos \ z cos fj&amp;gt; + y cos v, 

 2a# 2 = (a Pe) cos A. + 2 cos i*&amp;gt; y cos v, 

 with similar expressions for 26^, 260 2 &amp;gt; &c. 



Introducing these values into 



U e = 0, 



we obtain a result which may be written in the form 



where A, B, C contain eosX, cos/i, cos v in the second degree, and where 

 x , y , z enter linearly into B and in the second degree into G. 



Hence we see that on any straight line in space there will be in general 

 two screws belonging to any quadratic five-system. For the straight line 

 being given x , y , z are given, and so are cos X, cos //., cos v. The equation 

 just written gives two values for a pitch which will comply with the 

 necessary conditions. 



If we consider p e and also x , y , z as given, and if we substitute for 

 cos X, cos /j,, cos v the expressions x af y y y , z z respectively, we obtain 

 the equation of a cone of the second degree. Thus we learn that for each 

 given pitch any point in space may be the vertex of a cone of the second 

 degree such that the generators of the cone when they have received the 

 given pitch are screws belonging to a given quadratic five-system. 



If the equation 



be satisfied, then the straight lines which satisfy this condition will be 

 singular, inasmuch as each contains but a single screw belonging to the 

 quadratic five-system. As cos \, cos /j,, cos v enter to the fourth degree into 

 this equation it appears that each point in space is the vertex of a cone of 

 the fourth degree, the generators of which when proper pitches are assigned 

 to them will be singular screws of the quadratic five-system. 



If we regard cos X, cos p, cos v as given quantities in the equation 



then this will represent a quadric surface inasmuch as x , y , z enter to the 

 second degree. This quadric is the locus of those singular screws of the 

 quadratic five-system which are parallel to a given direction. Hence the 

 equation must represent a cylinder. 



