238 THE THEORY OF SCREWS. [225, 



If B = the two roots of the equation in p e will be equal, but with 

 opposite signs; as cos X, cos/i, cosy enter to the second degree in B it 

 follows that through any point in space as vertex a cone of the second degree 

 can be drawn such that each generator of this cone when the proper pitch is 

 assigned to it will equally belong to the quadratic five-system, whether that 

 pitch be positive or negative. 



If B = and (7 = 0, then both values of p e must be zero. Regarding 

 as , y , z as fixed, each of these equations will correspond to a cone with 

 vertex at x , y , z \ these cones will have four common generators, and hence 

 we see that through any point in space four straight lines can in general be 

 drawn such that with the pitch zero but not with any other pitch, these 

 screws will be members of a given quadratic five- system. 



226. Polar Screws. 



The general discussion of the quadratic screw-systems is a subject of 

 interest both geometrical and physical. We shall here be content with a 

 few propositions which are of fundamental importance. 



Let as before 



U e = Q 



be the homogeneous relation between the co-ordinates 1 ,...,0 n+1 of the 

 screws which constitute a quadratic ?i-system. 



Let 77 and denote any two screws other than 6 and chosen from the 

 enclosing w-system, from which the screws of the quadratic n-system are 

 selected by the aid of the condition U 6 = 0. If then we adopt the fertile 

 method of investigation introduced by Joachimsthal, we shall substitute in 

 mU e = for 6 1} ..., 6 n+1 the respective values 



The result will be 



TT &amp;lt;* r, 



where U* = -^- + ... + l+1 -- . 



C&quot;7i ar) n+l 



Solving this quadratic equation for I -f- m we obtain two values of this 

 ratio and hence ( 119) we deduce the following theorem. 



Any cylindroid of a given (n + l)-system will possess generally two screws 

 belonging to every quadratic n-system which the given (n + \)-system 

 encloses. 



If the two screws 77 and had been so selected that they satisfied the 

 condition 



