56 PENCILS OF LINES. 
these subgroups has the point O and some other point, as A, 
for the invariant points. For o/ positions of the point Q the 
resulting perspective transformations leave the point A as 
well as O invariant. It is easy to see that these ~! positions 
of Q must all be on the line AA’, because the second invari- 
ant point of a perspective transformation is found by drop- 
ping a perpendicular from @ on OX. Thus the~? perspective 
transformations obtained by taking the center of perspective 
at all points on a line perpendicular to OX form a one-par- 
ameter subgroup of G,(O). 
$8. Geometric Theory of Projective Transfor- 
mations of Pencils of Lines. 
We pass now to the geometric construction of projective 
transformation in other one-dimensional forms, viz.: in a 
pencil of lines through a point and a pencil of planes through 
a line. 
70. A Simple Construction. A perfectly obvious construc- 
tion for a projective transformation of a pencil of rays is as 
follows: Let two planes Pl and Pl meet in a line /; and let 
O, a point on /, be the common vertex of two pencils of rays, 
one in Pl and the other on Pl. The planes determined by 
three pairs of corresponding lines of the two pencils together 
with Pl and P7 determine a cone of the second order having 
its vertex at O and touching both Pl and Pl. Tangent 
planes to this cone cut Pl and P! in corresponding lines of 
the two pencils. If now Pl be revolved about l until it coin- 
cides with Pl, a projective transformation of the pencil in Pl 
is completed. The properties of this transformation and 
groups of such transformations may easily be developed by 
this method. 
71. Another Method. A second method for constructing a 
projective transformation of a pencil of lines is obtained by 
considering the process dualistic to that used for a range of 
