102 THEORY OF COLLINEATIONS. 
In Fig. 18 let the coordinates of P be (x, y) and of P, be 
(x, y,); let the coordinates of the three invariant points be 
(A, B), (A’, B’), and (A”, BY). 
Expressing the areas of the triangles in the last article in 
determinant form in terms of the coordinates of their vertices 
we have 
|v YW ‘| Le] 1 ce ye 22)| eae yal 
|A B 1| Angee | Ais ail |A BY 
HAV ISIE ii A” BY 1} Ales etl) ABU eh 
[| ea en ay Lee ie 
PAN BO a AUR BE Ki AIPRORL ALE!) 4d 
[Al BUS AUB ial AU BU ati| AB i 
1 yi 1 mil || 
AB: AB | 
Al BY Si Ala Bi eT) 
ja yp; et OP al 
AB a AB 1 
PAUSE ANB 
These three forms are not independent and the last may be 
regarded as superfluous. 
Putting k, =k and a =k’ then the most convenient form 
b 
is as follows: 
dB aH cee 2), ll ai i IL iO aoil|| 
A B 1 | Be AB yt Al UB i 
AU BUT PA TB al Al UBL ee Jes ah 
SE Sify Eee aq ay (10) 
\ixr Yr 2 Tee AOfing 3 | v1 Yi | LEY ia 
Al BY AL ESS a A!’ B' i AN Sse 
| A” BUS AUR 1| ALT BE th AES 
These implicit normal forms are capable of another inter- 
pretation ; the values of the determinants are proportional to 
