104 THEORY OF COLLINEATIONS. 
If we pass from Cartesian to homogeneous coordinates, 
these forms may be written: 
| a yz 0 | (ay eueraneO | 
SBCA |) 6) 2 ie eae 
Pt =\a Bp cra |> PYUr=\|a B oO RB |} 
| All Bl cl ki A” | | All Bi Cc" k/ B" 
| y Zz 0 | 
eee 
per =| BoC ko |: (12) 
| 4” BY c” wc"! 
Making the C’s and 2’s unity in (12) and dividing the first 
and second by the third, we return to equations (11). 
The law of formation of these determinants is evident. 
The determinant of the invariant triangle is bordered above 
by x, y, 2, on the side by A, k A’, k’ A”, ete. 
THEOREM 21. A collineation of type I can be expressed in a 
symmetric determinant form in which the coefficients are functions 
only of the eight natural parameters of 7. 
These explicit normal forms will be of great use to us in 
the following chapters. By giving to A, B, k, ete., the proper 
values any assigned collineation of type I can be written down 
at once. The analogy of these normal forms with the normal 
forms of type I in one dimension is evident (see Art. 17). 
131. Inverse of Tin Normal Form. If equations (10) be 
solved for x and y instead of «, and y,, we get the normal form 
of the inverse of 7. From the implicit normal form of T, 
equations (10), we see that the explicit normal forms of T 
and its inverse 7” differ only in the fact that k and k’ are 
changed into k~ and k’~’. 
