34 



Proceedings of the Royal Irish Academy. 



moment vanish. If AOA' exceeds 90°, then the Imes AB and AB' (fig. 7), 

 having passed the critical point, have become left-handed instead of right- 

 handed. By reflection from a looking-glass, a right-handed pair of lines will 

 appear as a left-handed pair, or vice versa. 



We have now to prove two general properties of a cylindroid, which may 

 be thus stated. 



The generator which contains the .screw of greatest pitch on the cylindroid 

 makes a left-handed pair with every other generator on the surface. 



The generator which contains the screw of least pitcli on the cylindroid 

 makes a right-handed pair with every other generator on the surface. 



We have from tlic formula (23) 



= = (Pa -Pfi) sine COS 0. (27) 



^8 Q Q- 



Fto. 7. — Right-lianded. Fio. 8. — Lofl-liandod. 



If J'a>Pfi a»d ^ ^l- 90°, then 2 is positive : hence, observing tlie 

 direction in which is measured (fig. 6), and that z is above the plane of 

 the paper, it is obvious that the generator defined by z and is above the 

 plane of the paper ; and from the position of this generator is plainly left- 

 handed with regard to a, for which ^ = 0. 



If, however, ^ 90'^, then the corresponding screw is below the plane 

 of the paper ; but it is still left-handed with regard to a. 



Thus the generator of the cylindroid which contains o is left-handed 

 with regard to every other generator on tlie surface. Of course, as a and /3 

 intersect at right angles, that particular pair are both right-handed and left- 

 handed. 



In like manner it can be shown that the generator of the cylindroid 

 which contains /3, the screw of least pitch on the surface, is right-handed 

 with regard to every other generator on the surface. 



