174 THE THEORY OF SCREWS. [174, 



The centres of all the hyperboloids coincide with the intersection of the 

 three principal planes of the cylindroids. It will be convenient to call this 

 point the centre of the three-system. 



We hence see that whenever three screws of a three-system are given, 

 the centre of the system is determined as the intersection of the principal 

 planes of the three cylindroids denned by each pair of screws taken suc 

 cessively. 



We may also show that not only are the family of hyperboloids concentric, 

 but that they have also their three principal axes coincident in direction and 

 situation with the principal axes of the pitch quadric. 



Draw any principal axis z of the pitch quadric. Two screws of zero 

 pitch belonging to the system will be intersected by z and we draw the 

 cylindroid through these two screws. Let L l and L.&amp;gt; be the two screws of 

 equal pitch p on this cylindroid. Let be the centre of the cylindroid, this 

 same point being also the centre of the pitch quadric, and therefore as 

 shown above of every p-pitch hyperboloid S p . As the centre bisects every 

 diameter, it follows that the plane OL 2 cuts the hyperboloid S p again in a 

 ray LI which is perpendicular to z and crosses L l at its intersection with z. 

 The plane containing L l and Z/ is therefore a tangent to 8 P at the point 

 where the plane is cut by z. As z is perpendicular to this plane it follows 

 that the diameter is perpendicular to its conjugate plane. Hence z is a 

 principal axis of S p , and the required theorem is proved. 



Let now S denote a screw system of the third order, where a, /3, y are 

 the three screws of the system on the principal axes of the pitch quadric. 

 Dimmish the pitches of all the screws of S by any magnitude k. Then the 

 quadric 



must be the locus of screws of zero pitch in the altered system, and therefore 

 of pitch + k in the original system ( 110). 



Regarding & as a variable parameter, the equation just written represents 

 tlie family of quadrics which constitute the screw system S and the reciprocal 

 screw system 8 . Thus all the generators of one system on each quadric, 

 with pitch + k, constitute screws about which the body, with three degrees 

 of freedom, can be twisted ; while all the generators of the other system, 

 with pitch k, constitute screws, wrenches about which would be neutralized 

 by the reaction of the constraints. 



For the quadric to be a real surface it is plain that k must be greater 

 than the least, and less than the greatest of the three quantities p a ,pp, p y - 



