NORMAL FORMS. ett 
139. Normal Form of Type II. A collineation of type II 
is the limiting form of one of type I when all three vertices 
of the invariant triangle approach coincidence, say at A, 
and all three sides of the same triangle approach coincidence 
in the line/. To find the normal form of the equations of 
type III, we let A” and A’ approach A along the curve s of 
Fig. 20. When the points A, A’, A” are consecutive points 
on the curve we may replace the coordinates A’, B’, A’, B’’ 
by the following expressions: A’=A+dA, B’/=B-+dB, 
A” =A+2dA+@A, B’=B+2dB+d°B. Weshall di- 
vide each determinant of the normal form of type I by twice 
the area of the invariant triangle AA’A’”. Wemay write 
2D= AA’. A’'A” sing. But when 4A, A’, A” are consecutive 
points we have chord A A’ = are AA’=ds, chord A’A” = are 
A’'A” =ds and sino = =cds, where c is the curvature of s 
at A. Hence 2D=cds’. 
In each determinant of the normal form of type I subtract 
the second row from the third, subtract twice the third row 
from the fourth and add the second row to the remainder, 
divide the new third row through by ds and the new fourth 
row by cds*. Substituting the above values of A’, etc., we 
find for the value of x, 
x y 1 0 
A B 1 
dA dB (k=1) | aA 
ds ds ds ‘ds 
dA @B ki —2k+1 kW —-kdA kid?A 
== 9 A +2 Sain 
cds? cds? cds? cds ds cds? 
t= ’ 
x ] iO) 
A B 1A 
dA dB, k-1 
ds ds ds 
aA dB i ki'—2k+1 
cds? cds? cds? 
and a similar expression for y,. 
