30 THE THEORY OF SCREWS. [25-27 



25. Screw Reciprocal to Five Screws. 



The determination of a screw reciprocal to five given screws must in 

 general admit of only a finite number of solutions, because the number of 

 conditions to be fulfilled is the same as the number of disposable constants. 

 It is very important to observe that this number must be unity. For if 

 two screws could be found which fulfilled the necessary conditions, then these 

 conditions would be equally fulfilled by every screw on the cylindroid 

 determined by those screws ( 22), and therefore the number of solutions of 

 the problem would not be finite. 



The construction of the screw whose existence is thus demonstrated, can 

 be effected by the results of the last article. Take any four of the five 

 screws, and draw the reciprocal cylindroid which must contain the required 

 screw. Any other set of four will give a different cylindroid, which also 

 contains the required screw. These cylindroids must therefore intersect in 

 the single screw, which is reciprocal to the five given screws. 



26. Screw upon a Cylindroid Reciprocal to a Given Screw. 



Let e be the given screw, and let X, p, v, p be any four screws reciprocal 

 to the cylindroid ; then the single screw 77, which is reciprocal to the five 

 screws e, X, //., v, p, must lie on the cylindroid because it is reciprocal to 

 X, /A, v, p, and therefore 77 is the screw required. 



The solution must generally be unique, for if a second screw were reciprocal 

 to e, then the whole cylindroid would be reciprocal to e ; but this is not the 

 case unless e fulfil certain conditions ( 22). 



27. Properties of the Cylindroid*. 



We enunciate here a few properties of the cylindroid for which the writer 

 is principally indebted to that accomplished geometer the late Dr Casey. 



The ellipse in which a tangent plane cuts the cylindroid has a circle for 

 its projection on a plane perpendicular to the nodal line, and the radius of the 

 circle is the minor axis of the ellipse. 



The difference of the squares of the axes of the ellipse is constant 

 wherever the tangent plane be situated. 



The minor axes of all the ellipses lie in the same plane. 



The line joining the points in which the ellipse is cut by two screws of 

 equal pitch on the cylindroid is parallel to the major axis. 



The line joining the points in which the ellipse is cut by two intersecting 

 screws on the cylindroid is parallel to the minor axis. 



* For some remarkable quaternion investigations into &quot; the close connexion between the 

 theory of linear vector functions and the theory of screws &quot; see Professor C. J. Jolj , Trans. Royal 

 Irish Acad., Vol. xxx. Part xvi. (1895), and also Proc. Royal Irish Acad., Third Series, Vol. v. 

 No. 1, p. 73(1897). 



