NEWSON: NORMAL FORMS OF PROJECTIVE TRANSFORMATIONS, au 
TYPE Ill. 
The invariant figure of type III consists of a single point A(a,b) 
and a line through it, as Al. Let P(x,y) be transformed to 
P,(x,,y,). The normal form of this type is given by the following 
equations: 
Ka Ya 1 hs yo 
a. |o) iy ia lo 
ie Ol, |m=1 oO 
= La (6) 
Se fe as PP aa 1| 
alloy al ee |e) a 
i jah toy! aes aba o| 
oe EV are 
lay be | 
I I m-1 oO) ‘ 
——— ka | = a® +ha. (6a) 
aya Sw Vow Kew Vad 
asses art lot Tl am beep 
if 00) it. 59a) _ (0) tial © 
The meanings of these constants or essential parameters are as 
follows: (a,b) are the coordinates of the point A; m=tan6, where 
6 is the angle which Al makes with the axis of x. The transforma- 
tion leaves invariant a pencil of conics all having contact of the 
third order at A; k is the reciprocal of the common radius of cur- 
vature at A of these conics, and hlk is the cotangent of the angle 
which the line of centres of these conics make with Al; a——cot- 
(x’,1), x’ being the ray through A which is transformed into the 
perpendicular at Al. 
TYPE Iv. 
Type IV represents a perspective transformation in which the 
centre A(a,b) is not on the axis of invariant points y—mx-tc. 
Let P(x,y) be transformed to P,(x,,y,). Two conditions are to 
be satisfied; the first is that P, P,, and A are collinear, the second 
condition is that the anharmonic ratio of the range (ABPP, )—k, 
where B is any point on the axis. 
The first condition is expressed by the equation 
x, a x—a (7) 
The second condition leads to the equation 
4 7 | 7 | 
en we Pasian 
ae | pen ae Peg tt 
i Im © (emmeanell fey 
sak : (7a) 
Soe Yee! See 
ON ear i Oe het) air 
