180 THE THEORY OF SCREWS. [178, 



The quadric surface whose equation is 

 (p a - k) x- + (pp - k) y- + (p y - k) z- + (p* - k) (p? - k) (p y - fc) = 0, 



touches the plane Px + Qy + Rz + S = 0, when the following condition is 

 satisfied : 



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 

 system, and, of course, two reciprocal screws, lie in the plane. 



From this it follows that all the screws of the system parallel to a plane 

 must in general 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 system; but this plane cannot contain any other screws; 

 therefore, all the screws parallel to a given plane must lie upon the same 

 cylindroid. 



179. Determination of a Cylindroid. 



We now propose to solve the following problem : Given a plane, deter 

 mine the cylindroid which contains all the screws, selected from a screw 

 system of the third order, which are parallel to that plane. 



Draw through 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 ( 174). 



Fig. 37. 



Let T l} T 2 (Fig. 37) be two points in which the two quadrics of constant 

 pitch touch the plane of the paper, which may be regarded as any plane 

 parallel to A ; then P is the intersection of the pair of screws belonging 

 to the system PT l} PT 2 , which lie in that plane, and P is the intersection 

 of the pair of reciprocal screws P R lt P R belonging to the reciprocal 



