58] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 51 



If the amplitudes a and /3 had opposite signs, then the point / should have 

 divided AB externally in the given ratio. 



57. Screw Co-ordinates. 



We have developed in the last chapter the general conception of 

 Screw Co-ordinates. In the case of the cylindroid, the co-ordinates of any 

 screw X, with respect to two standard scresvs A and B, are found by resolving 

 a wrench of unit intensity on X into its two components on A and B. These 

 components are said to be the co-ordinates of the screw. If we denote the 

 co-ordinates of X by X l and X 2 , we have 



^BX = AX 



l ~AB ~~ AB 



The co ordinates satisfy the identical relation, 



where e denotes the angle between the two screws of reference, that is, the 

 angle subtended by the chord AB. 



58. Reciprocal Screws. 



Every screw A on the cylindroid has one other reciprocal screw B tying 

 also on the cylindroid ( 26). Denoting as usual A and B by their corre 

 sponding points on the circle, we may enunciate the following theorem : 



The chord joining a pair of reciprocal screws passes through the pole of 

 the axis of pitch. 



The condition that two screws shall be reciprocal is 

 (p a -I- pft) cos 6 d af } sin = 0, 



where p a and p^ are the pitches, 6 is the angle between the two screws, 

 and d^ their shortest distance. It is easy to show that this condition 

 is fulfilled for any two screws A and B (Fig. 10), whose chord passes through 

 0, the pole of the axis of pitch PQ. 



42 



