NEWSON: NOAMAL FORMS OF PROJECTIVE TRANSFORMATION S. I29 
Since the determinants in (4) stand for double the areas of certain 
triangles we have 
_AP,AC. APAC AP,AB_ APAB 
— —— ¢ <c-1— St - : 
PAE BC NEEC’  oeREEEC APEC 
The normal form (4) may be expressed in terms of homogeneous 
coordinates by filling out the determinants with z, c, c, and cg. 
TYPE Il. 
The invariant figure of this type consists of two points A(a,b) 
and B(a,,b,), the line AB, and another line ACC,. Let the point 
P(x,y) be transformed to P,(x,,y,)- This transformation is com- 
pletely determined by the invariant figure and two essential para- 
meters k and a, where k is the anharmonic ratio of the pencil 
A(CBPP,) and a is given by xt = La. 
1 
The equations of the lines CA, BA, PA, P,A are respectively 
i Se shi oe v= 
Y=mxX-+ .. Jp a = NK ala VS —X+..., 
a,—a Xx—a 
y,—b., 
Vey Se 
X,;—a 
Whence we have 
y,—b y—b 
=m - —m 
X,—a X—a 
=k 
y,—b b,—b y—b b,—b 
X,—a a,—a Xx—a  a,—a 
This form reduces to 
ki Ya 2 x I 
ah ALT ee la est 
eee Os eek — iO 
bse eyaieare Kye hd 
jel Pameadtiee an fete ito. 
(aruebrs Sie (8 abn abr, 
The equations of the lines C,A, P,B, and PB are written 
Y=mX+b—am, 5 hii Io alec Gera el GS esse 
X4—a, X1—a, 
su sicra ee at a,(y—b,)+b,(« an) 
xXx—a, X—a 
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