38 Proceedings of the Royal Irish Academy. 



These theorems ought indeed to have been incorporated with the 

 earliest parts of the Theory of Screws; but I never noticed them until 

 1909. 



We can now see how to construct a system of pitches on the cylindroid 

 when the ruled surface merely has been given. 



Let 2m be the length of the axis of the cylindroid, and let p^ be any 

 linear magnitude, positive or negative. 



One generator, but only one, can be found on the surface which is right- 

 handed with regard to every other generator on the surface. To this generator 

 we attribute the pitch p,, - m. 



One generator, but only one, can also lie found on the surface which is 

 left-handed with regard to every otlier generator on the surface. To this 

 generator we attribute the pitch ^„ + in. 



The two generators thus indicated fonn the two principal screws on the 

 cylindroid ; and the pitch of any other screw on the surface whicli makes an 

 angle with the screw of maximum pitch will be 



(p, + w) cos'fl + (po - "i) siu'0. 



We may here notice the following extension of the theorems just given to 

 the case of the 3-.9ystem : — 



The screw of smallest pitch in a S-system forms a right-handed pair with 

 respect to every other screw of the S-system, and the screw of greatest pitch 

 in the S-system forms a left-handed pair with respect to every other screw of 

 the 3-sy8teni. 



Let (1), (2), (3) be the three principal screws of the S-system where 



Let be any other screw of the 3-system, and 0,, Q,, 0, its coordinates 

 with respect Lo the three principal screws. 



The two wrenches 0,", 0," compound into a single wrench on a screw ^ 

 lying on the cylindroid (12), and therefore cutting (3) at right angles. 



It is obvious that ij, and (3) are the principal screws on the cyhndroid 

 (<l>, (3j), and that must lie on this cylindroid. As p, is the smallest pitch 

 on any screw of the system, we must have p^> Pi', hence (3) is right-handed 

 Willi regard to every screw on this cylindroid among which 0, and therefore 0, 

 is included. 



In like manner, if ;/, be any screw on the cylindroid (2), (3), we have 6 

 a screw on the cylindroid (1), \p ; and as ^, is the greatest pitch of the 

 3-system, we must have the screws (1) and 0, forming a left-handed pair. 

 Thus the required theorem has been proved. 



As an illustration of various principles in this section, we may obtain the 



