Egan — Linear Complex, and a certain class of Twisted Curves. 65 



58. The equations of the types (a) and (b) can he wiitten 



ar,/f = x^jt^ = x^lt^ = X, ; (a) 



Xi/t^ = Xi/t* = cCi/t = Xi ; (b) 

 (c) can be written 



' x,/t' = x,/(t' + at') = x,/t- = aV(l + ct), 



Xi being the osculating plane at the cusp (f = 0, Xi = x^ = x, = 0), and Xi 

 that at an inflexion {t = co, x. = a-s = a'l = 0). 



If we express the fact that the tangents belong to a linear complex, we 

 find that the complex must be j^'i = 5ps3, and that this requires that 

 c = 5«. Hence we can write (putting t/a for f), 



iC,/t' = X,/(t' + 2t') = X,/t' = Xi/{1 + Qt). (c) 



In the same way we find that the curves of type (d) can be written 

 a;lt' = x,/t' = Xs/{t + 6f-) = .rj{l + 5t). (d) 



It will be noticed that if we identify all the homographic transformations 

 of a curve, each of the four types we have discussed reduces to a single 

 curve. 



The type (e) does not so reduce. "We shall consider one curve of this 

 type in detail in section X. 



IX. — Asymptotic Lines of Euled Surfaces belonging to a Linear 



CONGKUENCE. 



59. Picard has shown (loc. cit., section i) that on a ruled surface whose 

 generators belong to a linear complex there lies in general one curve of the 

 complex, cutting each generator twice. The osculating plane at any point 

 will be the tangent plane of the surface : the curve will therefore be an 

 asymptotic line. 



60. If the surface has two linear du'ectrices, the generators belong to a 

 linear congruence, and each asymptotic line will belong to a complex of the 

 congruence. These lines have been studied by Pittarelli {Bendiconti clella 

 E. Accademia dei Zincei, Sem. 2, 1894). I propose to supplement his work 

 by examining the relation of these lines to the pinch-points and the cuspidal 

 generators of the surface. 



61. Let the directrices be (2)^3 = Xt = 0, and (2') x, = Xz = 0. The 

 congruence is |j,2 = psi =0. A generator of the surface will be 



-» = -./(^)|. (28) 



■vs = xi(j)(6)\ 



R.I. A. PROC, VOL. XXIX., SECT. A. [9] 



