26 RECIPROCAL SCREWS. 



sary conditions, then these conditions would be equally 

 fulfilled by every screw on the cylindroid determined by 

 those screws ( 24), 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 cylin- 

 droid, which also contains the required screw. These 

 cylindroids must therefore intersect in the single screw,, 

 which is reciprocal to the five given screws. 



28. Screw upon a Cylindroid Reciprocal to a Given 

 Screw. Let e be the given screw, and let X, ju, v, p be any 

 four screws reciprocal to the cylindroid ; then the single 

 screw TJ, which is reciprocal to the five screws t, A, /i, i, p y 

 must lie on the cylindroid because it is reciprocal to 

 A, ju, v y p, and therefore r\ is the screw required. 



The solution must be unique, for if a second screw 

 were reciprocal to f, then the whole cylindroid would be- 

 reciprocal to c ; but this is not the case unless i fulfil cer- 

 tain conditions (24). 



29. Properties of the Cylindroid. We add here a few 

 properties of the cylindroid for which the writer is prin- 

 cipally indebted to his friend Dr. Casey. 



The ellipse in which a tangent plane cuts the cylin- 

 droid has a circle for its projection on a plane perpendi- 

 cular 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. 



