165] THE GEOMETRY OF THE CYLINDROID. 165 



reciprocal to that through P ; therefore the third intersection of S with the 

 surface must coalesce with P, or, in other words, S must be a tangent to the 

 surface at P. We have thus shown that in the case where two reciprocal 

 chords through a point on the cubic coalesce into one, that one must be the 

 tangent to the cubic at P. 



But the two reciprocal chords through a point will only coalesce when 

 the point lies on the hyperbola, in which case the two chords unite into the 

 tangent to the hyperbola. Consider, then, the case where the hyperbola 

 meets the cubic at a point P, inasmuch as P lies on the hyperbola, the two 

 chords coalesce into a tangent thereto, but because they do coalesce, this line 

 must needs be also a tangent to the cubic ; hence, whenever the hyperbola 

 meets the cubic the two curves must have a common tangent. Altogether 

 the curves meet in six points, which unite into three pairs, thus giving the 

 required triple contact between the hyperbola and the cubic. 



If a constant h be added to all the pitches of the screws on a cylindroid, 

 then, as is well known, the screws so altered still represent a possible cylin 

 droid ( 18). The variations of h produce no alteration in the cubic section 

 of the cylindroid ; but, of course, the hyperbola just considered varies with 

 each change of h. In every case, however, it has the triple contact, and 

 there is also a fixed tangent which must touch every hyperbola. This is 

 the chord joining the two principal screws on the cylindroid ; for, as these 

 are reciprocal, notwithstanding any augmentation to the pitches, their chord 

 must always touch the hyperbola. The system of hyperbolae, corresponding 

 to the variations of h, is thus concisely represented ; they must all touch 

 this fixed line, and have triple contact with a fixed curve : that is, they must 

 each fulfil four conditions, leaving one more disposable quantity for the 

 complete definition of a conic. See Appendix, note 4. 



We write the tangent to the hyperbola or the reciprocal chord in the 

 form 



L cos 2i/r + M sin 2-f + N = 0. 



If a pair of values can be found for x and y, which will simultaneously satisfy 



L = 0, M = 0, N = 0, 

 then every chord of the type 



L cos 2ir 



must pass through this point. The condition for this is, that the discriminant 

 of the hyperbola is zero, and we find the discriminant to be 



{m*h sin (X, 2 - 1) (h tan ft + m sin 2a). 



