122 DYNAMICS OF A RIGID BODY. 



Regarding k as a variable parameter, the equation just 

 written represents the family of quadrics which constitute 

 the screw complex S and the reciprocal screw complex 

 S'. Thus all the generators of one system on each qua- 

 dric, with pitch + k, constitute screws about which the 

 body, with three degrees of freedom, can be twisted ; 

 while all the generators of the other system, with pitch 

 - k y constitute screws, wrenches about which would be 

 neutralized by the reaction of the constraints. 



For the quadric to be a real surface it is plain that k 

 must be greater than the least, and less than the greatest 

 of the three quantities / a , p^ p y . Hence the pitches of 

 all the real screws of the screw complex S are inter- 

 mediate between the greatest and least of the three 

 quantities p n p^p r 



112. Screws through, a Given Point. We shall now 

 show that three screws belonging to S, and also three 

 screws belonging to S', can be drawn through any point 

 3/, y, z'. Substitute ^/, _/, 2', in the equation of the last 

 article, and we find a cubic for k. This shows that three 

 quadrics of the system can be drawn through each point 

 of space. The three tangent planes at the point each 

 contain two generators, one belonging to S, and the 

 other to S'. It 'may be noticed that these three tangent 

 planes intersect in a straight line. 



Two intersecting screws can only be reciprocal if 

 they be at right angles, or if the sum of their pitches be 

 zero. It is hence easy to see that, if a sphere be de- 

 scribed around any point as centre, the three screws 

 belonging to *$*, which pass through the point, intersect 

 the sphere in the vertices of a spherical triangle which 

 is the polar of the triangle similarly formed by the 

 lines belonging to S'. 



We shall now show that one screw belonging to S 



