126 



DYNAMICS OF A RIGID BODY. 



whence it follows that two values of k can be found, or 

 that two quadrics can be made to touch the plane, and 

 that, therefore, two screws of the complex, and, of course, 

 two reciprocal screws, lie in the plane. 



From this it follows that all the screws of the com- 

 plex parallel to a plane must lie upon a cylindroid. 

 For, take any two screws parallel to the plane, and 

 draw a cylindroid through these screws. Now, this 

 cylindroid will be cut by any plane parallel to the given 

 plane in two screws, which must belong to the complex ; 

 but this plane cannot contain any other screws ; there- 

 fore, all the screws parallel to a given plane must lie 

 upon the same cylindroid. 



114. Determination of the Cylindroid. We now pro- 

 pose to solve the following problem : Given a plane, 

 determine the cylindroid which contains all the screws, 

 selected from a screw complex of the third order, which 

 are parallel to that plane. 



Draw through O the centre of the pitch quadric a 

 plane A parallel to the given plane. We shall first 

 show that the centre of the cylindroid required lies in A. 



Fig. 3- 



Let TI, T z (Fig. 3) be two points in which the two 

 quadrics of constant pitch touch the plane of the paper, 



