36 



Proceedings of the Royal Irish Acadenvj. 



If, however, the two bounding screws of the cylindroid had been CD' 

 and A'S", it is plain that LLi and J/J/j must tend in directions opposite 

 to those given in the figure. In this case the pitch of the screw on XY 

 must be negative. "We thus obtain the general result that has been already 

 otherwise proved on p. Si, \\i. : — 



The screw of smallest pitch on the cylindroid makes a right-handed pair 

 with everj' other screw on the surface. The screw of largest pitch on the 

 cylindioid makes a left-handed pair with every other sci-ew on the surface. 



This theorem may be generalized in the following way, which will show 

 how any one generator on the cylindroid is related as to right-handedness 

 or left-handedness to all the remaining generators. 



Let APP'P" (fig. 10; be the circular representation* of the several screws 

 on a cylindroid. ML is the axis of piteh, and A is the point corresponding 

 to the screw of greatest pitch on the surface. Let P be a point on the circle 

 corresponding to any other screw on the cylindroid. 



A 



p 

 A 



Fio. 10. 



M L 



Fio. 11. 



Draw PP' (fig. 10) perpendicular to ML, and draw the diameter PP". 

 It follows from the principles of the circular representation that P' 

 corresponds to that screw on the cylindroid which intersects the screw 

 corresponding to P, while P" corresponds to the screw on the cylindioid 

 which is perpendicular to P. It is plain that P will have a relation as to 

 right-handedness or left-handedness with every other screw on the cylindroid 

 except the two critical screws P' and P" when the relation vanishes because 

 P and P" intersect, and P and P" are at right angles. 



Let ff, Khe the bounding screws of the cylindroid. We shall suppose 



rrcatiw." p. 4-7 and p. 120. 



