DYNAMICS OF A RIGID BODY. 129 



situation of a screw 6 belonging to a screw complex of 

 the third order of which the direction is given. The con- 

 struction is as follows : Draw through O the centre of 

 the pitch quadric a radius vector OR parallel to the 

 given direction of 9, 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 6. For the section in which A cuts the 

 pitch quadric has for a tangent at R a line perpendicu- 

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

 the section, and hence ( 114) one of the two screws of 

 the complex in the plane A must be parallel to OR. 

 It remains to find the actual situation of in the 

 plane A . 



Since the direction of is known, its pitch is deter- 

 minate, 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 

 0. This quadric must be intersected by the plane A in 

 two parallel lines. One of these lines is the required resi- 

 dence of the screw 0, while the other line, with a pitch 

 equal in magnitude to that of 0, but opposite in sign, 

 belonging, as it does, to one of the other system of 

 generators, is a screw reciprocal to the system. 



A family of quadric surfaces of constant pitch have 

 the same planes of circular section, and therefore every 

 plane through the centre cuts the quadrics in a system 

 of conies having the same directions of axes. 



The cylindroid which contains all the screws of the 

 screw complex parallel to one of the planes of circu- 

 lar section must be composed of screws of equal pitch. 

 A cylindroid in this case reduces to a plane pencil 

 of rays passing through a point. We thus have two 

 points situated upon the primary axis of the pitch quadric,. 



K 



