SCREW CO-ORDINATES. 37 



If two screws, 6 and 0, upon the cylindroid, are reci- 

 procal, then (m being the defining angle of 0), 



2/i (ai cos / + j3i sin /) (<n cos m + /3j sin m) = o, 

 or, p a cos / cos m + p$ sin / sin m = o. 



Comparing this with 20, we have the following useful 

 theorem : 



Any two reciprocal screws on a cylindroid are parallel 

 to conjugate diameters* of the pitch conic. 



Since the sum or difference of the squares of two con- 

 jugate diameters is constant f according as the pitch conic 

 is an Ellipse or a Hyperbola, we see that the sum or 

 difference of the reciprocals of the pitches of two reci- 

 procal screws on a cylindroid is constant according 

 as the pitch conic is an Ellipse or a Hyperbola. 



* Salmon, Conic Sections, 3rd Ed., page 129. 

 t Do. page 153. 



