30 Proceedings of the Royal Irish Academy. 



must be true for eaeli screw on the cylindroid, it is obvious that the 

 angle subtended by P must be constant, or, in other words, that the 

 locus of P is a circle. We are thus led to study the correspondence of 

 the screws on a cylindroid with the points on a circle. 



We can exhibit this correspondence in a very dii'ect manner by the 

 formula given in " Screws," p. 1 5. The position of a screw is defined 

 by the two equations 



y = X tan I, 



% = {^fa. - i'/s) sin I cos I ; 



while the pitch ^j of the screw is 



p = ])o. cos^ / + p^ sin^ I. 



In these equations ^a andji?j3 are the pitches of the two principal screws 

 of the cylindroid which intersect at right angles in the origin, and I is 

 the angle made by a variable screw with the axis of x. If we write 



i?o = i (i?a + p^) ; tn = + (i?a - Pi) ; 



and if we eliminate / from the two last equations, we obtain the result 



{p -Pof + s^ = m'^: 



regarding p and z as current co-ordinates, this equation denotes the 

 circle which corresponds point to screw with the screws on the cylin- 

 droid. It will, I thiak, be found that the correspondence presents the 

 various problems which arise with a degree of elegance hardly to be 

 anticipated from the mode in which the circle has been determined. 



Describe the circle HPQ, with ^as centre and m as radius, and the 

 plane representation of the cylindroid is complete. 



Draw an arbitrary straight line ZM (Fig. 1), and a perpendicular, 

 ffiV", to this line ; the distance jS^IT is equal to po, which is the con- 

 stant part in the expression of the pitch of a screw on the cylindroid, 

 the expression of any pitch being 



2h + ^}^ COS2 I. 



Take a point P, corresponding to any screw on the cylindroid, and 

 let fall PA perpendicular on ZJlf; then PA is the pitch of the screw 

 corresponding to P. If Q be a second point on the circle, then QB is 

 the pitch corresponding to Q, while the intercept AB between the 

 two perpendiculars is the shortest distance between the two con'e- 

 sponding screws. The angle subtended by the arc PQ, at any point 

 of the circumference, is the angle between the two screws. It will 

 now be easily seen how the screws determine the cii'cle just as they 

 determine the cylindroid. Draw a pair of parallels whose perpen- 

 dicular distance AB is equal to the shortest pei-pendicular distance 



