Egan— Zmear Complex, and a certain class of Twisted Curves. 67 



Cuspidal Generators. 

 63. If we have, when is near Q„, 



/=/o + {9-^rc{Q), .^ = 0„ + {e-e,rR{d), 



where G and JI are regular near 0„ and do not vanish for 6 = d„, the generator 

 9 = Of, is a cuspidal line of the surface. In effect, if we put - d„ = i, then, 

 for points on the surface near the line 0^,, 



X., -/o«i = «! {At' + BP + ...), Xs - (pgXi = 'Xi{A't- + ...); 



and the section of the surface by the plane x^ = kxi has a cusp where t = 0, 

 i.e. where the plane meets the generator 6q. It is easy to see that the 

 hyperboloid A'xi{x2 -foXi) = Axiijv, - (p„Xi) touches the surface along the 

 cuspidal line. 



Again, for the asymptotic line c, 



\' = c^'/f = c{M, + M,t + ...) {t = e- 0o). 

 Hence 



X^±c\a vU + cf + ...), <l,-<p, = A't' + B't^ + ..., f-f, = At- + Bt' + ... 



Hence X = Xifxi = ± c^{a + ht + . . .), 



r= {=<h-M)/x, = + ci{A'T + ...), 



Z = {xs- ^tfOi)lxi = A'f + . . . 



The tangent line is Y = Z = Q (the cuspidal generator) ; there are two 

 points of contact (^ = ± ac^). Hence each asymptotic line is hitangent to the 

 cuspidal generator : the i^airs of points of contact and the pairs of osculating 

 planes form involutions whose foci are the points and the planes common to the 

 cuspidal line and the directrices* 



The results in the more general case where 



f=fo+{0- OofG, f = ^, +(d- dofH 

 are easily worked out, but need not detain us here. 



Other Stationary Points. 



64. These will lie by twos on certain generators. Expressing that the 

 tangent plane to the surface has four-point contact with the curve, and 

 eliminating A, A', A" by equation (30), we find 



/"// - ¥'W = ¥"lt>' - H''^; (32) 



which gives these generators. 



* Pittarelli [loc. cit.) appears to confuse the case of pinch-points with that of cuspidal generators. 

 He says (if I have understood him correctly) that the 'singular generators' are hitangent to the 

 asymptotic lines, the points of contact lying on the directrices. 



[9*] 



