THE CYLINDROID. 19 



be all infinite, the general property reduces to the con- 

 struction for the composition of two translations or of 

 two couples. 



19. Form of the Cylindroid. The equation of the 

 surface only contains the single parameter pa. pp y 

 consequently all cylindroids are similar surfaces only 

 differing in absolute magnitude. 



The curved portion of the surface is contained be- 

 tween the two parallel planes z = + (p a - pp} y but it is to 

 be observed that the nodal line x = o, y = o, also lies upon 

 the surface. 



The intersection of the nodal line with a plane is a 

 double point (connode) or a conjugate point (acnode) 

 upon the curve in which the plane is cut by the cylin- 

 droid according as the point does lie or does not lie 

 between the two bounding planes. 



A model of a portion of the cylindroid is represented 

 in the frontispiece. In order to realize from the model 

 the actual form of the surface, the diameter of the central 

 cylinder must be conceived to be evanescent, and the 

 radiating wires must be extended to infinity. 



20. The Pitch Conic. Besides being acquainted with 

 the form of the cylindroid, it is also very useful to 

 have a clear view of the distribution of pitch upon the 

 screws contained on the surface. The surface being 

 given, one arbitrary element must be further specified 

 before that distribution is known. If, however, two screws 

 be given, then both the surface and the distribution are 

 determined. Any constant quantity may be added to 

 all the pitches of a certain distribution, and the distribu- 

 tion thus modified is still a possible one. 



Let p e be the pitch of a screw on the cylindroid 

 which makes an angle / with the axis of x\ then ( 1 1) 



p9 = fa cos 2 / + p ft sin'V. 

 C 2 



