138 THEORY OF COLLINEATIONS. 
2, 3, and 4 hold only for collineations of type I unless other- 
wise expressly stated. 
171. Resultant of Two Collineations in Cartesian Form. 
Let T and T, be given by the Cartesian equations 
F a¢r+by+eai _ aatbey tee | 
T EY Fh a par ? 1 asa+bsy+tes ? (8) 
and 
a an+tAy+tn _ am + fay + 72 
1 Ty ay = 
= 2 5 
fs an+ fsyitys ” ie 4301+ fsyitys © 
The first of these transforms the point (a, y) into (%,, y;) ; 
the second transforms (,, y,) into (x, y.). The resultant of 
T and T, is a transformation of same kind that transforms © 
(w, y) directly into (x., y,). The equations of this resultant 
are obtained by eliminating «, and y, from the equations of T 
and T, They are as follows: 
— (a4 +42 +asi) &+ (br41+ bef +b3n) y+ (aiai+ez/Ai+ s/n) 
oe (a1 43+ 4233+ as7/2) ©+ (6143+ b2/33+ bss) y + (c1 43+ €2/33-+ e373) ? 
i: (9) 
ips (a1 42+ 2 /72+ 3/2) «+ (bi 42+ be 32+ b372) y+ (c142+ €2/32 + Cafe) 
(b= (a1 43 + a9/73+ as7s) © + (b1 43+ b2/%3+ bs7’s) y + (C1934 ¢2/33+ e373) * 
The equations are again of the same form as (8) and the 
transformation T, is therefore a collineation. 
THEOREM 2. The resultant of two collineations is again a col- 
lineation. 
172. Determinant of the Resultant. The determinant of 
TAS 
lar4+a2n+as71 brat+befitbs aateAtesn | A; By, Cy 
Aj= | 0142+ 02 %+as7%2 61 424+ b2/32+bs7/2 cdo + co 92+ €372| == | Ar Bo Co ( 10) 
| 143+ a@2/33+as7/s bia3+bei3+bs/s  ¢1 43+ c233+ cs/s | As B3 C3 
This is equal to the product of the determinants of JT and T,, 
VAvA 3 
a b a aq fi fa 
A = |a2 be cl and A,= |# f& 1. 
as bs cs| ag fis Fs 
huss Ay — AAS 
It is evident that the determinant of T_, the resultant of 
