48 THE THEORY OF SCREWS. [52- 



screw can be ascertained from the position of its corresponding point on the 

 circle. 



Let us, for instance, seek the shortest distance between the two screws 

 A and J5. Since all screws intersect the nodal axis of the cylindroid at 

 right angles, the required shortest distance is simply the difference between 

 the values of m sin 20 for the two screws : this is, of course, the difference 

 of their abscissae, i.e. the length PQ. Hence we have the following theorem : 



The shortest distance between two screws, A and B, is equal to the pro 

 jection of the chord AB on the axis of pitch. 



We thus see that every screw A on the cylindroid must be intersected 

 by another screw A, and the chord A A is, of course, perpendicular to 

 the axis of pitch. The ray through S, parallel to the axis of pitch, will give 

 two screws, L and M. These are the bounding screws of the cylindroid, and 

 in each a pair of intersecting screws have become coincident. The two 

 principal screws, U and V, lying on a diameter perpendicular to the axis of 

 pitch, must also intersect. 



If all the pitches be reduced by p , then the pitch axis passes through 

 the centre of the circle, and the case assumes a simple type. The extremities 

 of a chord perpendicular to the axis of pitch define screws of equal and 

 opposite pitches, and every pair of such screws must intersect. The screws 

 of zero pitch will then be the bounding screws, while the two principal 

 screws will have pitches +m and m, respectively. 



53. The Angle between two Screws. 



This important function also admits of simple representation by the 

 corresponding circle. Let A, B (fig. 7) denote the two screws; then, if 6 

 and 6 be the angles corresponding to A and B, 



AST =26; B8T=28 , 



whence ASB = 2(8- 6 }. 



If IT be any point on the circle, then 



AHB = 0-8 , 

 and we deduce the following theorem : 



The angle between two screws is equal to the angle subtended in the circle 

 by their chord. 



The extremities of a diameter denote a pair of screws at right angles: 

 thus, A , in fig. 7, is the one screw on the cylindroid which is at right angles 

 to A. The principal screws, f/&quot;aud V, are also seen to be at right angles. 



