NEWSON: REAL COLLINEATIONS. 35 



Gc- one-parameter subgroups. For each of these groups r and s hate 

 fixed vahies while k is the variable parameter. 



Invariant curves and surfaces of hG\ {^ABCD)v%. — The one- 

 parameter group hGi (ABCD),s leaves invariant, besides the tetra- 

 hedron ( ABCD), a system of path curves which are usually curves 

 of double curvature and certain systems of surfaces on which are sit- 

 uated the invariant curves. In order to show this more clearly let us 

 consider the effect on a single point P anywhere in space of all the 

 transformations of the group Gi. Each transformation of the group 

 transforms the point P to some other point P,n. Since the cc^ trans- 

 formations in the group form a continuous system, there are cc^ of 

 these points Pm which form a continuous curve, viz., the path curve 

 of the point P. 



If we consider in this way the effect of the transformations of the 

 group on all the points of any arbitrary plane, we see that each point 

 of the plane traces a curve. Thus there are cc"^ of these path curves 

 invariant under all the transformations of Gi. Since our group con- 

 tains all the pseudo-transformations, corresponding to the values 

 k^O and k=Go, it follows that our path curves all pass through two 

 vertices of the invariant tetrahedron, but not through the other two. 



If a surface S be made to pass through oc^ of these path curves in 

 such a way that every point on each of these oci path curves lies on 

 S and also so that every point on S belongs to one of these path 

 curves, then such a surface is an invariant surface of the group Gi. 

 We can best determine these invariant surfaces by resorting to ana- 

 lytic methods. 



Equations of invariant surfaces of hG\ {ABCD)v^. — Let the in- 

 variant tetrahedron (ABCD) be the tetrahedron of reference and 

 let T be a transformation of the group Gi which transforms a point 

 P whose coordinates are (x, y, z, w) to Pi whose coordinates are 

 xi, yi, zi, wi. Pass planes through CDP and CDPi; let these cut 

 AB in Q and Qi. Then we have the cross-ratio ( ABQQi)=:k ; hence 

 B^ : B^'=k. Using proportional quantities, we have 



t^v-k. (1) 



In like manner we have the equations 



i7^i- = k- (2) 



f:--T=^"' (3) 



17:^ = 1^™ ii) 



?::f=k'-', (5) 



?f^T=t'-"-'. (6) 



