182 THE THEORY OF SCREWS. [179- 



drawn through T parallel to the external and internal bisectors of the angle 

 between the asymptotes are the two rectangular screws of the cylindroid. 

 Thus the problem of finding the cylindroid is completely solved. 



It is easily seen that each cylindroid touches each of the quadrics in two 

 points. 



We may also note that a screw of the system perpendicular to the plane 

 passes through T. Thus given any cylindroid of the system the position of 

 the screw of the system parallel to the axis of the cylindroid is determined*. 



180. Miscellaneous Remarks. 



We are now in a position to determine the actual situation of a screw 

 belonging to a screw system of the third order of which the direction is 

 given. The construction is as follows : Draw through the centre of the 

 pitch quadric a radius vector OR parallel to the given direction of 6, and 

 cutting the pitch quadric in R. Draw a tangent plane to the pitch quadric 

 in R. Then the plane A through OR, of which the intersection with the 

 tangent plane is perpendicular to OR, is the plane which contains 0. For 

 the section in which A cuts the pitch quadric has for a tangent at .R a 

 line perpendicular to OR; hence the line OR is a principal axis of the 

 section, and hence (179) one of the two screws of the system in the plane 

 A must be parallel to OR. It remains to find the actual situation of 6 in 

 the plane A. 



Since the direction of is known, its pitch is determinate, because it 

 is inversely proportional to the square of OR. Hence the quadric can be 

 constructed, which is the locus of all the screws which have the same pitch 

 as 6. This quadric must be intersected by the plane A in two parallel 



* In a letter (10 April 1899) Professor C. Joly writes as follows : Any plane through the 

 origin contains one pair of screws A and B belonging to the system intersecting at right angles 

 and another pair A and B belonging to the reciprocal system. The group A, B, A , B form 

 a rectangle of which the origin is the centre. The feet of the perpendiculars from on A and 

 on B and the point of intersection of A and B will lie on the Steiner s quartic 



(b-c)-y-z z + (c-a) 2 z-x- + (a-b) 2 x z y 2 - +(b-c) (c-a) (a-b)xyz. 



The point of intersection of A and B and the feet of the perpendiculars on A and B will lie on 

 the new Steiner s quartic 



(6 - c) 2 ?/ 2 z 2 + (c - ) z*-x*+(a - &) 2 zy= - (b - c) (c - a) (a - b) xyz. 



The locus of the feet of the perpendiculars on the screws of a three-system from any arbitrary 

 origin whatever is still a Steiner s quartic, but its three double lines are no longer mutually rect 

 angular. They are coincident with the three screws of the reciprocal three-system which passed 

 through the origin. This quartic is likewise the locus of the intersection of the pairs of screws 

 of the reciprocal system which are coplanar with the origin. There is a second Steiner s quartic 

 whose double lines coincide with the three screws of the given system which pass through the 

 origin and which is the locus of intersection of those pairs of screws of the given system which 

 lie in planes through the origin. It is also the locus of the feet of perpendiculars on the screws 

 of the reciprocal system. 



