[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 165 



Therefore 



t/i = -^ [2ci (f/o- UU,) + c,U + c,\, 



V U 



V, = -^[2c,{V.-VV,)-c^V + c,]. 



On introducing these values of Ui and Vi in (24), we find the general 

 solution of the Normal System in the form 



(24') (^=) = 2fi(f/2F3 + U,V,) + 2ci 



r c/2- t/c/3 , 



— , — an 



JU' 



+ 2fi 



^' ^^' rfz' + C3(I/2-F2)+C4(t/3+y3)+C2. 



n/F' 



Thus we arrive at the following fundamental set of solutions of the 

 Normal Svstem: 



x = {U2V,+ U,V,) + 





JIP 



F2-FF3 , 



— , dv , 



(26) y=U2-V2, z=U3+V3, w = l. 

 If we introduce the substitution 



u=U3,v=Vs, U2=U', F2=F', 

 the equations (26) become (on dropping the bars) 



(27) x = 4:{U+V)-2{u-v) (U'-V), 

 y=U' —V , z = u-\-v, w = \. 



It remains to find the equations of the complexes C'{v) and C"{u) 

 whea expressed in the Cartesian coordinates {x, y, z). We shall omit 

 the details of the calculation involved in passing from the moving 

 tetrahedron (3^, z, p, a) to the fixed trihedral (x, y, z). 



The equations of the complexes C'{v) and C"{u) turn out to be 



(28) C'{v): 2{y5z)-{8x)-\-4v{ôy)-\-4:V'{ôz)=0, 

 C"{u): 2{ybz) + {bx)+^u{by)-4.U'{bz)=i). 



The loci of the axes of C'{v) and C"{u) are the cylinders 



y = F', z = — V 

 and y = U', z = it 



respectively. We shall say that the complexes of a family are att ached 

 to a cylinder C when the axes of the complexes are generators of C. 

 The principal parameters of C'{v) and C"{n) are +i and -| respec- 

 tively. 



The results established above may be recapitulated in the following 

 theorem: 



Theorem. — // the Directrix Qiiadric of a surface S consists of two 

 coincident planes, the two families of linear complexes determined by the 

 asymptotic curves on S are attached to two cylinders with parallel genera- 



