128 KANSAS UNIVERSITY QUARTERLY. 
lines or through the invariant points; and that the product of these 
three anharmonic ratios when taken in the same order round the 
triangle is unity. These three ratios are designated by k,k-¢, and 
ke-l, 
Let the point P(x,y) be. transtormed to Pj (x,.y,)- Uheauhan 
monic ratio of the pencil C(ABPP,)=k. The equations of the 
lines AG, “BG PC, ary |C.aretrespectively: 
gy Derm 7.01) Gp SEP bapa UMN a aa 
a,—a elle kas 
A tees in 
X,—a, | > 
when X and Y are the current coordinates. Hence we have 
VarOe Dee Vase Oa Oe 
Kael ao xd ASSEly 
—-—k . 
Mince Os Dinas y—b, Diet Oe 
; X;—@y a,—a, X—aAg ay—Ay 
Simlarly the pencil B(CAPP, )——k-! Whence 
Hise Pos lice vy Ocala 
X,——-a, a—a, X—a, a—a, 
-—}c-1 : 
Yar Diy oe Deri) i Dh aren 
X,—a, Aa,—a, X—a, ay—a, 
These forms readily reduce to* 
Xeyie OL Koo yieee ae iv a 
Vaya Die al ayeloys | la, bies (a eae 
er [by St) a. |B. Sit la lpi lee Sloy) it 
= ke (4) 
| 
ee beh 1 Xa | KTV ala 
lag igen Fee oper ala Sal) ai Day el 
an 2D a al ost E| aie Dig i EVA oe 
This form (4) is analogous to the first normal form of (1); for 
the latter may be written 
Rs | ie 1| 
mm Im | 
—_—k = 
xg xq itl 
| Te) 20 
*In Continuierliche Gruppen, pages 22 and 76, Lie gives results which, if carried a step 
farther, would lead to these forms. 
