44 THE THEORY OF SCREWS, [49, 50 



In this form it i.s obvious that to increase each of the three quantities 

 Pa&amp;gt; pp&amp;gt; P by in does not affect the equation. This is of course the well-known 

 property of the cylindroid. ( 20.) 



If the quadratic have two equal roots, then all the screws of the cylindroid 

 form a plane pencil, and all have equal pitches. 



The discriminant of the quadratic is 



^ a +^) 2 -4w 0(j; a +p0)cos(a/S)-4^ a p0 sin 2 (a/S)+4OT^ 2 si 

 - (p a + p ft ) cos (a/3)} 3 + (p a + ppf sin 2 (a/3) - 4&amp;gt;p a p ft sin 2 (a/ 

 (^a-^) 2 )sin 2 (a/3). 

 Hence discarding the case when (a/3) = we have 



d.fi=0, p*-p ft = 0, 



as was to be expected. 



