146 THEORY OF COLLINEATIONS. 
Boyes mA B C 0 
B C C BEG! kA’ kB! Ok 0 
So 2 1 1 Tel All Jel BY eu 
By Gi kecy |= |B" CY WAY BY KC om VII 
O  @ An CIR Cy Cy 
0 O An! By (Gil kexGi! 
OM 0. AW BY GE hel Gur 
[RAs Cs AL B (6; 0 
oh C C A! @ KA! kB’ k@’ 0 
2 2 | A” CURA" kB! kicv 0 
~ ee) 
Agi Ge! koCo l\=16 0 AM B, G ( |\0 VIII 
0 0 Aj! By! Cy k, Cy! 
0 0 UI SULA OFLA HUA OFLA 
(Ale AMES per 0 
3 AI B! TAL cB Vike! 0 
‘1 ie “ k a the A" BU RA" kB" kev 0 IX 
Ao! BR. i'l ~ We 0 A; B, C C 
OF 0 A’ By! Cy! kd; 
oO Av! Bl Gi kG 
These nine equations are not independent; dividing each of 
these equations through by any one of them, we obtain eight 
independent equations which enable us to express the eight 
parameters, A,, B,, k,, etc., in terms of A, B,k, A,, k,, ete. 
1793. Determinant of the Resultant. It was shown in Art. 
172, that, if 7 and T, are two collineations whose determi- 
nants are respectively A and A,, the determinant A, of T,, 
the resultant of 7 and T,, is equal to the product of A and 
A,; 4:s= AA. Making use of the value of the determinant 
of Tin Art. 132 we have the equation 
An, Bs Cy |3 AY BV 'C; |'5:" "AG By Cy i 
ke. ig,!| Ao! Bo! CG! | = kk’ k, lee Al Bl Gt} ANE men Callie x 
ACU Ba OnE A” BY Cl\ | Ay BY Gil 
180. The Group G,. We have shown in three different 
ways by the eliminations in Arts. 171, 175 and 178 that the 
resultant of two collineations is again a collineation. The 
inverse of a collineation is also a collineation as was shown 
in Art. 77, and again by the form of 7‘ in equations (12). 
Since the system of all the * collineations of the plane has 
both group properties, it is proved that they form a group Gy. 
