175] FREEDOM OF THE THIRD ORDER. 175 



Hence the pitches of all the real screws of the screw system S are inter 

 mediate between the greatest and least of the three quantities p a&amp;gt; pp, p y . 



175. Construction of a three-system from three given Screws. 



If a family of quadric surfaces have one pair of generators (which do not 

 intersect) in common, then the centre of the surface will be limited to a 

 certain locus. We may investigate this conveniently by generalizing the 

 question into the search for the locus of the pole of a fixed plane with 

 respect to the several quadrics. 



Let A be the given plane, / be the ray which joins the two points in 

 which the given pair of generators intersect A, X be the plane through / 

 and the first generator, Y the plane through / and the second generator, 

 B the plane through / which is the harmonic conjugate of A with respect 

 to X and Y. Then B is the required locus. 



For, draw any quadric through the two given generators, and let be 

 the pole of A with respect to that quadric. 



Draw a transversal through cutting the plane A in the point A l and 

 the first and second generators in X l and Y t respectively. Since A 1 is on 

 the polar of it follows that OZ^Fj is an harmonic section. But the 

 transversal must be cut harmonically by the pencil of planes I(BXAY) 

 and hence must lie in B, which proves the theorem. 



In the particular case when A is the plane at infinity, then is the 

 centre of the quadric. A plane parallel to the two generators cuts the 

 plane at infinity in the line /, and the planes X, Y and B must also contain 

 7. Then A, B, X, Y are parallel planes. Any transversal across X and Y 

 is cut harmonically by B and A, and as A is at infinity, the transversal must 

 be bisected at B. It thus appears that when a family of quadrics have one 

 pair of non-intersecting generators in common, then the plane which bisects 

 at right angles the shortest distance between these generators is the locus 

 of the centres of the quadrics. 



If therefore three generators of a quadric are given, the three planes 

 determined by each pair of the quadrics determine the centre by their 

 intersection. The construction of the axes of the quadric may be effected 

 geometrically in the following manner. Draw three transversals Q l} Q 2 , Q 3 

 across the three given generators R 1} R 2 , R 3 . Draw also two other trans 

 versals Hi, R 5 across Q 1} Q 2 , Q 3 . Construct the conic which passes through the 

 five points in which R 1} R 2 , R 3&amp;gt; R it R s intersect the plane at infinity. Find 

 the common conjugate triangle to this conic and to the circle which is the 

 intersection of every sphere with the plane at infinity. Then the three 



