132 KANSAS UNIVERSITY QUARTERLY. 
TYPE V. 
Type V represents a perspective transformation in which the 
centre is on the axis. The two conditions to be satisfied are that 
I I 
A, P, P, are collinear and that AP we ae 
The first condition leads to 
Vitae y=). 
a 8 
X,;—a x—a nO 
The second condition is satisfied by the equation 
I I 
— = +a. (8a) 
Pane 1 HY ac] 
tae (pS 20" Uidied diay edi 
T, 1G) |z- maso. 
REDUCTION TO CANONICAL FORMS. 
By a suitable choice of coordinate axes the five normal forms 
above may be reduced to their simplest or canonical forms. 
If equations (4) be made homogeneous and the invariant triangle 
be taken for triangle of reference we get 
eee and pa —jei* . 
Zs Z ie zZ 
if the side z be made the line at infinity, these reduce to 
Xj kes x, and iva ly (9) 
In like manner equations (5) and (5a) reduce to 
lee ade a +a; or x,=kx and y,=y-+ta. (10) 
Zs Zz eke 
Equations (6) and (6a) reduce to 
US an) eee Ka |-—q® ha. ‘ 
Macy SA a ee 
Interchanging y and z and then making z—1 we get 
X,—=x-++a and y,=y-+kex+ Sat ha. AGE 
Equations (7) and (7a) reduce to 
— : and zl =k2; whence we have x,—kx and y,=ky. (12) 
1 Z, 
Equations (8) and (8a) become 
