236 THE THEORY OF SCREWS. [225 



equated to zero would correspond to a three-system selected from the 

 enclosing four-system. We know (| 176) that three screws of a three-system 

 can be drawn through each point. We have, consequently, three screws 

 through the point for each of the two factors of U e , i.e. six screws in all. 



The equation of the 6th degree in p e contains also the co-ordinates x , y , z 

 in the sixth degree. Taking these as the current co-ordinates we may 

 regard this equation as expressing the family of surfaces which, taken 

 together, contain all the screws of the quadratic three-system. The screws 

 of this system which have the same pitch p e are thus seen to be ranged on 

 the generators of a ruled surface of the sixth degree. All these screws 

 belong of course to the enclosing four-system, and as they have the 

 same pitches, they must all intersect the same pair of screws on the 

 reciprocal cylindroid ( 212). It follows that each of these pitch surfaces of 

 the sixth degree must have inscribed upon it a pair of generators of the 

 reciprocal cylindroid. 



Ascending one step higher in the order of the enclosing system we see 

 that the quadratic four-system is composed of those screws whose co-ordinates 

 satisfy one linear homogeneous equation L = 0, and one homogeneous 

 equation of the second degree U Q. We may study these screws as 

 follows. 



Let the direction cosines of a screw 6 be cos X, cos /u,, cos v. If the 

 reference be made, as usual, to a set of canonical co-reciprocals we have 



cos X = B! + 6., ; cos //, = # :i + 64 ; cos v = 5 + # 6 . 



We therefore have for a point x , y 1 , z on 6 the equations ( 218) 

 2a0! = (a + p e } cos A, z cos JJL + y cos v , 

 2a0 2 = (a p e ) cos X + z cos /* y cos v, 



with similar expressions for 3 , # 4 , 5 , 6 fi . 



Substituting these expressions in L = and U = and eliminating p e , 

 we obtain an homogeneous equation of the fourth degree in cos X, 

 cos fi, cos v. If we substitute for these quantities x x, y y , z z , we 

 obtain the equation of the cone of screws which can be drawn through 

 x , y , z \ this cone is accordingly of the fourth degree. We verify this con 

 clusion by noticing that if U = were the product of two linear functions, 

 this cone would decompose into two cones of the second degree, as should 

 clearly be the case ( 218). 



It remains to consider the Quadratic Five-system. In this case the 

 enclosing system includes every screw in space, and the six co-ordinates of 



