204 THE THEORY OF SCREWS. [204, 



204. The Pitch Conies. 



The discussion in 203 will prepare us for the plane representation of 

 the screws of given pitch p, for we have 



P (0? + 0/ + 3 2 ) - M a - M 2 - M 2 = - 



This, of course, represents a conic section, and, accordingly, we have the 

 following theorem : 



The locus of points corresponding to screws of given pitch is a conic 

 section. 



A special case is that where the pitch is zero, in which case the locus is 

 given by 



This we shall often refer to as the conic of zero -pitch. 



Another important case is that where p is infinite, in which case the 

 equation is 



0* + 3 *+0 3 *=o. 



The conic of zero-pitch and the conic of infinite pitch intersect in four points, 

 and through these four points all the other conies must pass. The points, of 

 course, correspond to the screws of indeterminate pitch : we may call them 

 P,, P,, P,, 1\. 



Any conic through these four critical points will be a conic of equal- 

 pitch screws. 



As a straight line cuts a conic in two points, we see the well-known 

 theorem, that every cylindroid will contain two screws of each pitch. 



The two principal screws on a cylindroid are those of maximum and 

 minimum pitch ; they will be found by drawing through P l} P 2 , P 3 , P 4 , the 

 two conies touching the straight line corresponding to the cylindroid. The 

 two points of contact are the screws required. 



If a and ft are the two principal screws on a cylindroid, then any pair 

 of harmonic conjugates to a and ft represent a pair of screws of equal pitch. 



For if S + kS = be a system of conies, then it is well known that the 

 pairs of points in which a fixed ray is cut by this system form a system in 

 involution. The double points of this involution are the points of contact 

 of the two conies of the system which touch the line. 



205. The Angle between Two Screws. 



From the equations of the screw given in 201, we see that the direction 



