BREWSTER : COLLINEATIONS OF SPACE. 285 



leave this same figure invariant and form a one-parameter group. We 

 shall choose as our invariant surface one of those quadrics which is 

 tangent to p, the plane of the invariant points, and has this line as a 

 tangent line. This group we call 9 Gi(lmp^F). 



The generators 1 and m, in which the tangent plane cuts the 

 quadric, are rays of the invariant pencil in that plane. Therefore, 

 along each of them there is a parabolic transformation, differing from 

 each other only by a constant factor. Moreover, these two generators 

 are harmonically separated by the axis of the pencil of invariant 

 points ; for, in the general case, two reciprocal polars with regard to a 

 quadric surface are. harmonically separated by that surface. But 

 should those two polars, as a special case, intersect, they must do so 

 in a tangent plane, and their point of intersection is on the surface. 

 But the two axes referred to above form just this special case of re- 

 ciprocal polars ; and the lines harmonically separating them are the 

 generators cut from the quadric by the tangent plane. Hence, these 

 two axes are harmonically separated by the two invariant generators of 

 the surface. 



9. For each pair of generators we may have oo 1 positions of the 

 tangent line of invariant points, i. e., oo 1 of the one-parameter groups 

 just discussed. Hence, there are of space od 4 collineations of type IX 

 which leave invariant a given quadric surface, each of which leaves 

 also invariant one generator of each system on the quadric and a line 

 of invariant points tangent to the quadric at the intersection of these 

 two generators. 



10. Higher groups. — Consider next the resultant of two trans- 

 formations y T(lmp£F) and 9 Ti(lmp^F), where t and h denote different 

 positions of the line of invariant points in the tangent plane. Since 

 the generators 1 and m are always to be harmonically divided by the 

 axes of invariant planes and points, as t shifts in the tangent plane, 

 the axis of the pencil of planes must shift correspondingly in the 

 same plane. We have left in each case a system of quadrics through 

 a point tangent to a given plane. Therefore, in each component, the 

 quadric of which 1 and m are generators remains unchanged. The 

 resultant space collineation leaves invariant the given quadric and its 

 two generators, a new pencil of planes, a new line of points, and is 

 again of type IX. But the line t may take oo 1 positions in the plane 

 and through the given point. For each position there exists, as has 

 been shown (art. 8), a one-parameter group leaving invariant a sys- 

 tem of quadric surfaces. Each of these one-parameter groups has in 

 common the quadric tangent to the invariant plane along 1 and m. 

 Therefore, the resultant of two transformations 9 T(lmp^F) and 



