( 218 ) 



D{AE - F') r' + {AE J^ BD — C' — F') r' {Ez' — 2Fz-^A)-\- 



-{- B {Ez'' — 2Fz i- Ay = O (4) 



As this can be decomposed into two factors of the form 

 Lr^ -{- M {Ez^ — 2Fz -f- ^), th'i singular surface S consists of two 

 quadratic surfaces of revolution. 



These touch each other in the cyclic points /j and I^ of the plane 

 XOY and in the points B^ and B^ on OZ determined by 



Ez^ — 2Fz-\^A — {). 



The two surfaces cut each other according to the four isotropic 

 right lines indicated by the equations 



A-^ -^ y"" = Q and Ez"" — 2Fz -\- Az=:Q . . . . (5) 



If 52 is symmetric (C=0) the two parts of the singular surface 

 have as equations 



{AE — F') (.r' 4- if) + B {Ez^ — 2Fz-\-A) = 0, . . (6) 



D i^' + y') + Ez' — 2Fz -^ A = (7) 



If we find ^ = and D = 0, then S breaks up into the four 

 planes (5) and i2 is a particular tetraedal complex. 



Out of (3) it is easy to find that the cones of the complex of the 

 points B^, B^, /j and I^ break up into pencils of rays to be counted 

 double. 



These points shall be called hisingular. 



§ 3. The rays of the complex resting on a straight line / touch 

 a surface which is the locus of the vertices of the cones of the 

 complex touched by /. This axial surface is in general of order four 

 and of class four and possesses eight nodes. ^) 



We shall determine the axial surface of OZ. The points of inter- 

 section (0, 0, z') of an arbitrary cone of the complex with OZ are 

 indicated by the equation 



iE{x^ + y^) + Bl^ z'^ - 2 [F (.^'^ + y^) + Bz] z' + [A {.V^ + y^) ^ Bz^] = 0. 



This has two equal roots if 



{{AE - F^) (.^.^ -^y') + B {Ez- - 2 Fz ^ A)] {x^ -\-y') = ^ . (8) 



So the axial surface of OZ consists of the two isotropic planes 

 through the axis and a quadratic surface of revolution which might 

 be called the meridian surface. If i2 is symmetrical, it forms part 

 of the singidar surface as is proved out of (6). 



Also the axial surface of the right line /<» lying at infinity in 

 XOY breaks up into two planes, and a quadratic surface. Its 



1) Sturm, Liniengeometrie III, p. 3 and 6. 



