108 THEORY OF COLLINEATIONS. 
ee B 
cil through B we have t’= lim. —.. Hence | == lim. — 
B=0 sinB ah 2! AG: 
But in the triangle ABC, even when C is very near to 
Te _ sinA 
7 BC 
since AB is the limit of BC. 
° sin B sin A sin ¢ 
; and lim. —— =lim. —— = —~ 
A, we have - AG Na ae en 
f= Se 4, (14) 
THEOREM 25. In a collineation of type If the parabolic con- 
stants ¢ and ¢/ of the two one-dimensional parabolic transformations 
along the invariant line Al and through the invariant point B are 
. sin 
connected by the relation t= —— ¢’. 
137. Normal Form of Type II. The normal form of the 
equations of a collineation of type II may be readily obtained 
from those of type I by considering type II as the limiting 
form of type I when one of the invariant points, as C, ap- 
proaches A along the line b. In the normal form of I, equa- 
tions (11), subtract the second row from the fourth row in 
each determinant and then divide the fourth row in each de- 
terminant by d, the length of the segment AC. Now let C 
; AN — A os IB Tg 
approach A and put lim. — (5, Juan 7 =’, and 
A'vN=A d A’=A a 
ki —1 
lim. —— =t; we then get 
Lif 
le oO f @ | x y 1 O 
ABi1A | AB1 B 
| a’ Ba kA’ | | A’ Br kB | 
c ec 0 tA+c |e € © tB+c! | 
iy = -— a —, (15) 
| a: Th eh <8) Cas eee Lene O, 
[Aa Bid AB at ae 
jl TY i TD Al Bist aki 
kee AOE OG OD 
Evidently ¢ and c’ are cosine and sine of # the angle which 
l of the invariant figure of type II makes with the «-axis. 
If the axes of coordinates are oblique, then ¢ and c’ are pro- 
