202 THE THEORY OF SCREWS. [202, 



If we make 



* = (P*-pa)* , @ = (p 3 -pi)* , f Y = (pi-prf, 

 the equations of the four planes are expressed in the form 



+ ax + fty + yz a/3y = 0, 

 ax + fiy + jz afty = 0, 

 + ax /3y + yz a/3y 0, 

 + ax + (3y &amp;lt;yz afty = 0. 



It is remarkable that the three equations of the axis for each of these screws 

 here coalesce to a single one. The screw of indeterminate pitch is thus 

 limited, not to a line, but to a plane. The same may be said of each of 

 the other three screws of indeterminate pitch ; they also are each limited 

 to a plane found by giving variety of signs to the radicals in the equations 

 just written. We have thus discovered that the complete locus of the 

 screws of a three-system consists, not only of the family of quadrics, which 

 contain the screws of real or imaginary, but definite pitch, but that it also 

 contains a tetrahedron of four imaginary planes, each plane being the locus 

 of one of the four screws of indefinite pitch. 



203. Relation of the Four Planes to the Quadrics. 



The planes have an interesting geometrical connexion with the family of 

 quadrics, which we shall now develop. The first theorem to be proved is, 

 that each of the quadrics touches each of the planes. This is gcometrically 

 obvious, inasmuch as each quadric contains all the screws of the system 

 which have a given pitch p\ but each of the planes contains a system of 

 screws of every pitch, among which there must be one of pitch p. There 

 will thus be a ray in the plane, which is also a generator of the hyper- 

 boloid but this, of course, requires that the plane be a tangent to the 

 hyperboloid. 



It is easy to verify this by direct calculation. 

 Write the quadric, 



( Pi - p) a* + (p,- p) f + (p 3 -p)z~ + ( Pl - p) (p. 2 - p) (p 3 - p ) = 0. 



The tangent plane to this, at the point x, y , z, is 

 ( p, - p) xx + (p,-p) yij + (^ 3 _ p) zz &amp;gt; + ( pl 



If we identify this with the equation 



ax + fiy + ryz a/3y = 0, 



