Egan — Linear Complex, and a certain class of Ttvisted Curves. 59 



48. Value of the functions 



%ai {6) X, it) + 2a.- it) X, ((9) = {d- ty W, 



2n< {&) Xi (0 - 2a.. (t) Xi (6) = (6- ty r 



for an algebraic P-curve. 



We have seen that %ai(0)Xi(t) or 2)3;. T; is equal to iT/gL{t)yip{B). 

 Now ag is AL^(0)/(i>{6) ; hence 



^jiiXi = aZ(t) L{d) TTte af^, where a = Ai = const. 



Hence 



{0 - tyW = aL{t)L{d) {w,e<yf'^ + Tr^afJ), \ 



(6 -tyV = aZ{t) L (6) (tt,, ^fi - net afi) .] ^"*^ 



It follows, by the theorem of par. 38, that the necessary and sufficient con- 

 dition that the same linear complex should contain three consecutive tangents at 

 each of the two ordinary jjoints t and 6 on cm cdgebraic P-curve is that the jmnts 

 shoidd be connected by the relation VW = 0. 



49. The equations V = 0, W = define symmetric correspondences on 

 the curve, of orders (n - 3, n - 3) and (n - 6, n - 6) respectively. A 

 pair of conjugate points of either correspondence has the property just 

 stated; but there is certainly some geometric difference between the two 

 correspondences which I have been unable to discover. 



If the curve itself belongs to a linear complex, W vanishes identically, as 

 we saw ; while V only vanishes when the chord t6 is a line of the complex. 



The Self-Conjugate Points of V = and of W = 0. 



50. If we allow 6 to approach t, we find 



U V = - (03)/3, 

 e ^=^ t 



Lt jr- (06) + (60) 2 (.33) 

 9->< 6! 6! 



Hence the self -conjugate points of V are given by (03) = ; in other words, 

 they are the singularities of the curve, each occurring with the order of 

 multiplicity p-3 (sections iv and v). The self-conjugate points of W are 

 given by (33) = 0. 



Hence every ordinary (3, 2, 1) point at which six consecutive tangents belong 

 to a linear complex satisfies the equation (33) = 0. 



This explains why we found that the identity (33) = characterizes 



curves of a complex among P-eurves. 



[8*1 



