Ball — Contrihutions to the Theorij oj Screws. 37 



at first that F lies on the semicircle HAK\ as already shown, each pair of 

 screws are left-handed which lie in that region of the cylindroid defined 

 by HAK. 



Let X be a variable screw on the cylindiciid which we shall suppose to 

 move in succession over every generator on the surface ; and we shall follow 

 the relations as to right-handedness or left-handedness with the other screws. 

 So long as X is in HAK then /-" and X are a left-handed pair, and there will 

 be no breach of continuity in their condition until X, moving in the direction 

 of the arrow from P towards P', reaches P'. When X has attained this 

 position, the screw corresponding to P and the relation of right-handedness or 

 left-handedness vanish. Thus P' is a critical point, so that when X crosses 

 P' and enters P'P", PX becomes right-handed. This condition remains till 

 X reaches P", which is another critical point ; for there the two screws P 

 and X are at right angles. As X passes to the other side of P" into the 

 region P"H, it again forms a left-handed pair with P; and this condition 

 remains while X moves through the semicircle P"HAP and returns to P. 



Thus we see that in fig. 10 the screw corresponding to P makes a right- 

 handed pair with each of the screws on the cylindroid whose representative 

 points lie between P' and P", while, with every other generator on the surface, 

 P makes a left-handed pair. 



If the circumstances had been those represented in fig. 11, then the 

 screws corresponding to P and X would have formed a right-handed pair. 

 This condition would have continued as X advanced in the direction of 

 the arrowhead until the critical point P" was reached ; and PX would be a 

 left-handed pair so long as X was moving from P" to P'. At P' another 

 critical stage is passed ; and P and X would be a right-handed pair as X 

 moved round through P'KP. 



Thus we see that as fig. 11 is drawn the generator corresponding to P 

 makes a right-handed pair with every generator on the cylindroid except 

 those represented by the points on the arc P"P' . 



In both figures we see that P and any screw in the hatched portion of 

 the circumference form a left-handed pair, while P and any screw in the 

 cross-hatched portion form a right-handed pair. 



As a particular case, we note that, if P coincides with A (fig. 10), then 

 P' coincides with P" , the cross-hatched portion disappears ; and consequently 

 every screw on the cylindroid makes a left-handed pair with P. 



On the other hand, if P coincides with B (fig. 11), then P'P" vanishes; 

 and the circle is completely cross-hatched : hence we see that the screw of 

 least pitch on the cylindroid forms a right-handed pair with o\<'ry olliin- 

 screw on the cylindroid, 



