24 THE THEORY OF SCREWS. [17- 



17. Form of the Cylindroid in general. 



The equation of the surface contains only the single parameter p*pp, 

 consequently all cylindroids are similar surfaces differing only in absolute 

 magnitude. 



The curved portion of the surface is contained between the two parallel 

 planes z = (papp), but it is to be observed that the nodal line x = 0, y= 0, 

 also lies upon the surface. 



The intersection of the nodal line of the cylindroid with a plane is a 

 node or a conjugate point upon the curve in which the plane is cut by the 

 cylindroid according as the point does lie or does not lie between the two 

 bounding planes. 



18. The Pitch Conic. 



It is very useful to have a clear view of the distribution of pitch upon 

 the screws contained on the surface. The equation of the surface involves 

 only the difference of the pitches of the two principal screws and one arbitrary 

 element must be further specified. If, however, two screws be given, then 

 both the surface and the distribution are determined. Any constant added 

 to all the pitches of a certain distribution will give another possible distribu 

 tion for the same cylindroid. 



Let p g be the pitch of a screw 6 on the cylindroid which makes an angle I 

 with the axis of x ; then ( 13) 



Pe = Pa cos 2 1 +pp sin 2 1. 

 Draw in the plane x, y, the pitch conic 



where H is any constant ; then if r be the radius vector which makes an angle 

 I with the axis of x, we have 



H 

 P = -?&amp;gt; 



whence the pitch of each screw on a cylindroid is proportional to the inverse 

 square of the parallel diameter of the conic. 



This conic is known as the pitch conic. By its means the pitches of all 

 the screws on the cylindroid are determined. The asymptotes, real or 

 imaginary, are parallel to the two screws of zero pitch. 



19. Summary. 



We shall often have occasion to make use of the fundamental principles 

 demonstrated in this chapter, viz., 



