NEWSON : REAL COLLINEATIONS. 103 



faces invariant under the collineations of the group Gi(Apl) are ob- 

 tained by eliminating t from all pairs of equations formed from (1), 

 (2), and (3). From (1) and (2) we get: 



f(xi, yi, zi)=f(z, y, z)=|y2 + hyz — nxz — Cz2 = 0. I 



From (I) and (3), and making use of the identity in I, we get: 

 (xi, yi, zi, wi)=rf(z, y, z, w)=-3y^H ^ — y^z — mnxyz — gnyz- + 



n'wz 



Cz''=0. II 



From (2) and (3), and using the identities in I and II, we get: 



f(xi, yi, zi, wi) = f(x, y, z, w)=j x [J( hz + y ) + '^z] — ^ j + 



j x(gz + ky + ^^)-w(hz + y)| 



)f(gz+ky + ^^)-(hz + y) [^(hz + y)+^z]{-Cz^=0. Ill 



Making z = in equations (4), (5), and (6), the last disappears, and 

 the modified forms of (4) and (5) can be put into the form : 



'^— =— + 1 



\ ^^ ' (10) 



I ^ = ^ + mt-^ + ^t-^ + kt. 



Eliminating t from these, we have the following equation of the in- 

 variant conies in the plane p : 



f(xi, yi, wi)=f(x, y, w)=^x2 + kxy-yw-Cyl IV 



In I, II, III, and IV, C is the arbitrary parameter of the family of 

 surfaces. 



Equation I represents a system of quadric cones having A for a 

 common vertex, 1 for a common element, and p for a common tangent 

 plane. Equation II represents a family of cubic ruled surfaces having 

 1 for a common line and the plane p for an inflectional tangent plane 

 along the line 1. The curves of intersection of I and II are the path 

 curves of the group Gi(Apl). 



The two families of surfaces have the line 1 in common ; hence their 

 curves of intersection are of a lower degree than the sixth. Taking a 

 section of both surfaces by the plane w=0, we get the following sys- 

 tem of curves : 



|y'^ + hyz — nxz = Cz^ and ™y^ + "^~ " y^z — 



mnxyz — gnyz' = Cz^. (11) 



Eliminating x from these equations, we find three points of inter- 

 section exclusive of those on 1. Hence the intersections of the sys- 

 tems of surfaces I and II are oc- twisted cubics in space. They all 

 pass through A and have 1 for a common tangent at A. 



Theorem 4. The group G6(Apl) contains oo' one-parameter sub- 



