NEWSON: REAL COLLINEATIONS. 37 



section of these invariant surfaces are the invariant path curves of 

 the group Gi. 



The geometric meaning of r and s. — It is not difficult to determine 

 the geometric meaning of the two constants r and s. Any tangent to a 

 path curve in the plane ABC cuts the side of the triangle ABC in 

 three points which form with the point of contact a range of four 

 points whose cross-ratio is constant and equal to r. Any tangent to 

 a path curve in space cuts the three planes DAB, DAC, DBC in 

 three points which form with the point of contact a range of constant 

 cross-ratio r. For if path curves and tangent be projected from D 

 on the plane ABC we get the path curves and tangent in the plane 

 ABC with the usual meaning of r. 



In like manner s is seen to be the cross-ratio of the point of con- 

 tact of a tangent to a path curve and the three points where the 

 tangent cuts the three planes ABC, ABD, ACD. In general, a 

 tangent to any path curve in space cuts the faces of the invariant 

 tetrahedron (ABCD) in four points; these with the point of contact 

 of the tangent form a set of five points on a line ; r and s are two in- 

 dependent cross-ratios of these five points. All other cross-ratios 

 among these five points may be expressed in terms of r and s. Since 

 these five points are all real, r and s must both be real. 



Theorem 4. The constants r and s are two independent cross- 

 ratios among the range of five points in which a tangent to a path 

 curve cuts the tetrahedron (ABCD) and its point of contact. 



§2. Two-parameter Subgroups of hGs (ABCD). 



Having shown that the group hGg (ABCD) contains oc"^ one-para- 

 meter subgroups, it will be shown next that these one-parameter sub- 

 groups unite in certain instances to form two-parameter subgroups of 

 liGg (ABCD). It will be found that a two-parameter subgroui3 of 

 this kind is characterized by the fact that it leaves invariant one and 

 only one family of surfaces on which lie the path curves of its one- 

 parameter subgroups. 



Two-parameter groups leaving invar'iant a family of cones. — 

 If r remains constant while s assumes in turn all real values, we have 

 a system of oc^ one-parameter groups, all of which leave invariant the 

 system of cones given by equation (VII) ; for the equation of this 

 family of cones is independent of s. The other six systems of sur- 

 faces given by equations (I)-(VI) vary ass varies, and are not in- 

 variant under all the cc"^ transformations which leave the cones of 

 (VII) unchanged. This system of oc'^ transformations leaving a 

 family of cones invariant evidently forms a two-parameter group, 

 the parameters being k and s. There is one such group for every 

 value of r. 



