156 THE ROYAL SOCIETY OF CANADA 



linear complexes; and that, therefore, each of these surfaces has two 

 straight line directrices. We shall denote these by (ô/, So') and 

 (5i", Ô2") respectively. The directrices (5/, 82) constitute the two 

 branches of the flecnode curve of R' {i.e., the locus of points at which 

 tangents have fourth order contact), and these two branches of the 

 flecnode curve are coincident or distinct according as the Invariant d' 

 does or does not vanish.* A similar statement applies to the direc- 

 trices (5"i, ô"2) and the Invariant 6". 



It has also been shewn (section 4 of the paper cited above) that the 

 loci of (Ô1, §'2) and (5/', Ô2") are complementary reguli of a quadric 

 surface, called the Directrix Quadric.f 



It may happen that the Directrix Quadric degenerates into two 

 planes, distinct or coincident. From geometrical considerations it is 

 apparent that, when the Directrix Quadric degenerates into two 

 coincident planes, the two Invariants di, 6" must vanish. The con- 

 verse of this however is not easily proved by geometrical reasoning. 



In the succeeding sections we shall prove that when the Invariants 

 6', 6" vanish, the Directrix Quadric consists of two coincident planes 

 (of course, the Invariants Î2', fi" vanish) ; we shall also determine the 

 finite equations of the surface for this case. 



Normal Equations. 



We shall now determine the normal form of system (6) when the 

 Invariants Q', U", 6' , d" vanish. At the outset we may exclude the 

 case {a)' or (6) equal to zero; since, if either {a') or (Jb) vanish, the 

 surface 5 will be ruled, J and when the asymptotic curves of a ruled 

 surface belong to linear complexes the surface has straight line 

 directrices.§ 



From the conditions 



we deduce the equation 



S2'=0, fi"=0 



ôMog 

 whence (by integration) 



(j) 



(10) ^^'-^")^ eul -0' 



b V ' 



where U and V are functions of u alone and v alone respectively, and 

 where C/4=0, V^ 0. 



*Wilczynski, Projective Differential Geometry of Curves and Surfaces, Teubner, 

 Leipzig, 1906, p. 130. 

 ]S, §§3, 4, 7. 

 XMi, p. 293. 

 §5, p. 171. 



