( 221 ) 



When in equation (1) of tlie complex 52 the coefficients E and F 

 are zero, the complex ii. is displaced in itself by each helicoidal 

 movement with axis OZ. This complex can be called helicoidal. 



The singular surface has as equation 



{BD — C^){x^ -\- y"")^ AB^O; (21) 



so it consists of a cylinder of revolution and the double laid plane 

 at infinity. 



§ 6. By homographic transformation the complex i2 can be changed 

 into a quadratic complex with four real bisimjular points. 



If we take these as \'ertices of a tetrahedron of coordinates 

 O^OJJJJ^, it is not diflicult to show that the equation of such a complex 

 has the form 



Ap\, + Bp\^ + 2 Cp,,p,, + 2Dp,,p,, ^-2Ep,,p,, = 0. (22) 



If we again introduce the condition that the section of the* 

 cone of the complex with one of the coordinate planes consists of 

 two right lines we find after some reduction for the singular surface 



A{D-E)y,^y,'^^2\AB-{C-D){C-E)]y,y,y,y, + B{D-E)y,^y,:^ = . (23) 



So this consists of two quadratic surfaces, which have the four 

 right lines 0^0^, 0^0^, OJJi and <J.JJ^ in common. 

 For A = 0, B = the complex proves to be tetraedral. 

 For D = E the equation is reducible to 



^P\. + Bp\, + 2{C- D)p,,p,, = 0, 

 and indicates t\vo linear complexes. 



For the axial surfaces of the edges 0^0^ and O^U^ we lind 



,x,x,\2Ax,x,-\-{D-E).v,x,\ = () .... (24) 

 and 



' x,x,\2Bx,x,^{D-E)x,x,\^0 .... (25) 



For a point (0, t/^, 0, yj of the edge OJJ^ the cone of the complex 

 is represented by 



Ay.-^x,-^ -^r2{C-E)y,y,x,x,-\-By,^x^ = ^-,. . (26) 



so it consists of two planes through 0^0 ^. 



This proves that the edges 0^0^, 0^0^, OJJ^, OJJ^ are double 

 rays of the complex '). 



1) See Sturm, Liniengeometrie III, pp. 416 and 417. 



