Egan — Linear Complex, and a certain class of Twisted Curves. 71 



System of Curves and Surfaces derived from one of these Curves. 



72. We saw in section ix, par. 66, that a curve k is an asymptotic line 

 on a quartic scroll S whose parametric equations are 



■Jc^ = {fi - 2Yxi, Xi = {ix + 2yxi. 

 The equation of S is therefore 



^(x2/:r,) + y(x3/xt) = 4, 

 or, rationalizing and arranging according to descending powers of Xi and Xt, 

 256 Xi'Xi^ - 16 XiXi(xiX3 + x^x^ + {x^x^ - x^x^f = 0, 



which shows that the two sheets of the surface which touch along the cuspidal 

 line p touch the hyperboloid H=x^X3-x■iX^=0 along the line /). It is easy 

 to see that H contains, besides ^j, the two lines (± 1), (+ i). These we shall 

 call J and / respectively. 



73. Sis, the locus of the secant {t,t'^) of a curve k; in other words, it is 

 the locus of the join of two conjugate points in one of the two involutions 

 into which the 2, 2 relation breaks up. The corresponding line (t, - <"') 

 connected with the other involution generates a surface ;S". 



To investigate S', we put t' = it on the curve k : we have then to find the 

 locus of the line {f, l/t'). E"ow the 2, 2 relation is of the same form as before. 

 Hence, by taking as coordinate planes the osculating planes at t' =1, 0, oo, - 1, 

 we find the equations of the curve in the form (35), with t' for t, and of course 

 with a different tetrahedron of reference. The equation of S' will therefore be 

 the same as would be the equation of S if derived from equations (35). 



74. Both S' and ;S' have the bitangent of the curve as a cuspidal line. 

 Further, they touch along this line. In effect, the tangent plane to S is 

 determined by the hyperboloid K, which is determined by its three generators 

 of the same species p, I, J. Consider the corresponding lines p, I', J'. I' passes 

 through the points f =± i, i.e. t = ±l; it therefore coincides with ./, and 

 J' in like manner is /. Hence K and S' are identical, and the two surfaces 

 S, S' touch along the conrmon cuspidal line. 



Again, they touch along the curve k, since it is an asymptotic line on each. 

 Hence the intersection of S and S', which is of the sixteenth degree, is made 

 up of the curve counted twice and the bitangent counted six times. 



75. Starting with a curve k, we obtain two surfaces, S and S', having k as 

 a common asymptotic line. These surfaces contain an infinity of curves k 

 (their asj'mptotic lines) ; aU of these have the same bitangent p, and their 

 inflexions lie by twos on the lines / and J. Taking any such curve (say on S), 

 we obtain a new surface S', and a new set of eur\'es k. Proceeding thus we 



