RECIPROCAL SCREWS. 25 



droid. For, suppose that three screws A, ju, v, which were 

 reciprocal to the four given screws did not lie on the same 

 cylindroid, then any screw on the cylindroid (A, ju), and 

 any screw i// on the cylindroid (A, v) must also fulfil the 

 conditions, and so must also every screw on the cylindroid 

 (0, \/,) ( 4). We should thus have the screws reciprocal 

 to four given screws, limited not to one surface, but to a 

 family of surfaces, which is impossible. The construction 

 of the cylindroid which is the locus of all the screws re- 

 ciprocal to four given screws, may be effected in the fol- 

 lowing manner : 



Let a, |3, 7, S be the four screws, of which the pitches 

 are in descending order of magnitude. Draw the cylin- 

 droids (a, 7) and Q3, ). If or be a linear magnitude inter- 

 mediate between p ft and / 7 , it will be possible to choose 

 two screws of pitch <r on (a, 7), and also two screws of 

 pitch a on Q3, ). Draw the two transversals which in- 

 tersect the four screws thus selected ;* attribute to each 

 of these transversals the pitch - <r, and denote the screws 

 thus produced by 0, 0. Since intersecting screws are 

 reciprocal when the sum of their pitches is zero, it fol- 

 lows that 9 and must be reciprocal to the cylindroids 

 (a, 7) and (|3, S). Hence all the screws on the cylindroid 

 (0, 0) 'must be reciprocal to a, /3, 7, S, and thus the pro- 

 blem has been solved. 



27. Screw Reciprocal to Five Screws. The problem 

 >of the determination of a screw reciprocal to five given 

 screws must admit of a finite number of solutions, 

 because the number of conditions to be fulfilled is the 

 same as the number of disposable constants. Now it is 

 very important to observe that that number must be one. 

 For if two screws could be found which fulfilled the neces- 



* Two lines can be drawn which will intersect four non-intersecting lines. 

 -Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., page 426. 



