Two- AND THREE-PARAMETER GROUPS. 31 
37. Transformations of Pencils of Lines and Planes. The 
theory sketched in the foregoing pages applies equally well to 
the one-dimensional transformations of the lines of a flat pen- 
cil or the planes of an axial pencil. There are two varieties 
of such transformations, viz., those with two invariant ele- 
ments and those with only one invariant element. 
In the first case let O be the vertex of a flat pencil, A and 
A’ the two invariant lines of the pencil, and « and x, any pair 
of corresponding lines in the transformation. Then we have 
the cross-ratio O( A’Avx,) =k, and the theory requires no 
further development. 
The second case, with one invariant element, may be de- 
duced as the limiting form of the first case in the following 
manner: Let O( A’Axu,) =k; whence O(A’vAx,) =1—k. 
Writing out the last cross-ratio in full, we have: 
sin(AOA’) | sin (a1 0A’) 
sin (AOr) : sin (“10x ) ae 
Whence 
sin (a Ox) 1=k 
sin(AOx) . sin(#10A’)  sin( AOA! ) ; 
But (x,Ox) = (A’Ox) — (A’Ox,) ; therefore, 
lim sin( A’Ox)cos(A’Ox1) —cos( A/Ox)sin(A!Ox1) lim 1-k sas 
A'=A sin (AOx) . sin(#10A‘) an Al=A sin (AOA) WH § 
Hence cot (x,0A’) — cot(xOA’) =t, 
or cot 0, =cot0-+t. (36) 
THEOREM 19. In a transformation of a pencil of lines (or 
planes) of type II, the difference of the contagents of the angles 
made with the invariant line (or plane) by a pair of corresponding 
lines (or planes) is constant for all pairs of corresponding lines 
(or planes). 
