Normal Forms of Projective Transforma- 
tions. 
BY, Hie Ba eNEWiSONr 
A linear fractional transformation in one variable, viz: 
. __ax--b 
1 ex-Edi 
is always reducible to one or the other of the following normal 
forms:* 
Dg ue yao I I 
or = =O (Ge) 
X,;—n en X,—m! x—m!/ 
These two forms correspond to the two types of transformations of 
this kind, viz: transformations with two invariant points, and 
transformations with one invariant point. In these normal forms 
the constants of the transformations are the ‘essential parameters’; 
each constant has a definite meaning. 
In the first form m and n are the coordinates of the two invariant 
points and k is the anharmonic ratio of these invariant points and 
any pair of corresponding points x and x, in the transformation. 
In the second form m’ is the single invariant point and a is a con- 
stant which characterizes the transformation. If p and p, are cor- 
responding points, the normal form shows that 
I I 
w= ——_ — —_.. 
iNet 0). ean 
Some point q on the line is transformed to infinity; hence 
I 
m/q 
This linear fractional transformation may also be interpreted as 
a projective transformation of the lines of a flat pencil or of the 
planes of an axial pencil. The first type of transformation leaves 
two rays (planes) of the pencil invariant, and k is the anharmonic 
ratio of the pair of invariant rays (planes) and any pair of corres- 
*See Klein’s Elliptische Modulfunctionen, pp. 164 and 173. 
(125) KAN. UNIV. QUAR., VOL. VII, NO. 3, JULY, 1898, SERIES A. 
