20 THE CYLINDROID. 



Draw in the plane #, y, the conic 



where H is any constant ; and if r be the radius vector 

 which makes an angle / with the axis of x, we have 



&*?:> 



whence the pitch of each screw on a cylindroid is pro- 

 portional to the inverse square of the parallel diameter 

 of the pitch conic. 



Being given the cylindroid, we require further to 

 know the eccentricity of the pitch conic, and then the 

 pitches of all the screws are determined. 



21. Summary. It is one of the main objects of the 

 present essay to associate a geometrical conception with 

 the solution of each problem. To do this effectively we- 

 shall often have occasion to make use of the principle 

 demonstrated in this chapter, viz., 



That a cylindroid can be drawn so that not only shall 

 two of its generators coincide with any two given screws a 

 and j3, but that when all the generators of the surface become 

 screws by having pitches assigned to them according to the 

 law of distribution enunciated in 20, the pitches assigned 

 to the generators which coincide with a and ]3 shall be equal 

 to the given pitches of a and ]3. 



