( 217 ) 



aiid it can be distijigui.-hcd as a .ri/imnetric complex of revolution. 

 Til is is the case with the complexes of tangents of surfaces of 

 revol lit lull. 



We determine the general equation of the quadratic complexes of 

 revolution with axis OZ in the coordinates of ra^s 



Ih = ''^' — ■^*' ' Vi — V — V ' 2>3 = ^ — '^'. 



i\ — y~ — ^y' ' P5 = -•^■' — •^-' ' Pt — '^i/' — 3/'^'- 



Bj substitution of 



Pi = «Pi — i^^ ' Pa = /?i^ + «i^ » Ps = ^3' 



1\ = «P. — t^/'s ' Ps = /5/>4 + «i>6 ' Pe = Pa' 



(where «' -|- Z^'' = 1) in tlie general quadratic equation we easily 

 find that the equatioji of an 12 can contain terms only with 



ipi' -^P.'h (pi'iPs'h P3' Pe^ ipiPs—P.Pi) and (p^p^ -\-p,P,)- 

 As the latter combination can be replaced by — />3 ^^^ in consequence 

 of a wellknown identity- we liiid tVjr i2 the equation 



^(Pi'+P.=)-fi?P3^ + 2Q>3Pa+A^a^+^iP.^+P3^) + 2^T/>iPa-P,y^)=0. (1) 



If C=0, equation (1) does not change when x is replaced by 

 — X ; so it represents a symmetrical complex. 

 The coordinates of rays 



q^:=u — u' , q^ = v — v' , g^ =r k — w , 



q^ ■=! vw' — u'v' , ^5 zzz icic — riiv' , q^ z= uv' — vu' , 



where u, v and lu represent the coordinates of planes are connected 



with the coordinates p by tlie wellknown relations 



Pi'i4=P2'9ó=Pz'q,= Pa -qi^p.-q^— p, ■ q»- 



So i2 can also be represented by 



^q.'-^q^) + ^q^'+^Cq,q,+Bq,^^A{q,^-\-q,^)^2F{q,q,-q,q,) = 0. . (2) 



This equation is found out of (1) by exchanging p/c and qf;, and 

 of A, B, a D, E, F and E, I), C, B, A, — F. 



§ 2. The cone of the complex of the point {x\y\z') has as 

 equation : 



A{x-w'yJrA{y-y'y-\-B{z-z'y^2C{y'x-x'y){z-z')JrD{y'x~x'yy^ 

 -\-E{z'y-y'zy^E{z'x-xzy-^2F{x-x){x'z-z'x)^2F{y—y'){y'z-z'y)=0.{^) 



In order to find the equation of the singular surface we regard 

 the cones of the complex whose vertices lie in XOZ and note the 

 condition expressing that the section of such a cone and XOY 

 breaks up into two right lines. After suppression of the factor z^ 

 which is to be rejected and substitution of .ï' -}- ^' = r* for x"", we 

 find the equation 



