36 KANSAS UNIVERSITY QUARTERLY. 



Suppose that Pi is a fixed point and P a movable point depending 

 upon the jDarameter k. Eliminating k from equations (1) and (3), 

 we get 



^=^^=Const. 



W" Z Wl"^" Zl 



Clearing of fractions, we have 



x"-^ y=Cw'^^ z. (I) 



For different values of C this equation represents a system of invari- 

 ant surfaces through the path curves of the group. 



In like manner eliminating k from equations (2) and (6) we get 



y.-rs-l ^r^Q^r-rs-l ^r. ( JJ ) 



Also from ( 4 ) and ( 5 ) we get 



x'-i y'-'•s=Cz'-lw"^ ( III ) 



It should be noted in each of these cases we have eliminated k from 

 the values of the cross-ratios along ojiposite edges of the tetrahedron. 

 If we eliminate k from the cross-ratios along edges which lie in a 

 plane face of the tetrahedron, we obtain the equation of the plane 

 path curves which lie in that face. The systems of cones whose ver- 

 tices are the vertices of the invariant tetrahedron and whose bases are 

 the plane path curves of the opposite faces are also invariant surfaces 

 of the group Gi. Eliminating k from ( 2 ) and ( 3 ), we have 



y^-iz = Cx«, (IV) 



which is the equation of the system of invariant cones whose vertices 

 are at A. 



In like manner eliminating k from equations (3) and (6), (1) and 

 (4), (1) and (2), we get 



^r-l y 1-1-4 r.^Cw--*, (V) 



^v-rs-1 z = Ow'-'"*, (VI) 



x■^-ly = Cw^ (VII) 



which are the equations of the invariant cones whose vertices are re- 

 spectively B, C, and D. 



We have thus found seven systems of invariant surfaces of the 

 group Gi ; three of these systems are ruled surfaces which pass through 

 four edges of the invariant tetrahedron, and four of them are cones 

 which have their vertices at the vertices of the invariant tetrahedron. 

 The intersections of any two of these systems of surfaces give us the 

 path curves of the group Gi. 



Theorem 3. There are seven distinct families of ruled surfaces 

 invariant under all the transformations of the group liGi (ABCD)r3; 

 four of these families are families of cones. The oc"^ curves of inter- 



