174 THEORY OF COLLINEATIONS. 
This set of relations on the elements of M implies another re- 
lation, viz.: that the determinant of M is unity. This may 
be readily shown as follows: Form the symmetric determi- 
nant, 
iC aping 
D'=\p m7, 
lg rn 
and substitute for these elements their values on the left hand 
side of R,; the new determinant factors readily into A’ D’. 
Whence we have A’=1, where A is the determinant of VM; 
but by Art. 181 we can take only the positive sign. Hence 
A\ = il. 
If the six equations of FR, are multiplied respectively by 
x?, y*, 2°, 2uy, 2uz, 2yz, and added we get 
f= U(aya+biy+e.z) 2+m (arx+boy+erz)2+ ete. = lx?+my?+nz2+2pry+ 2quz+2ryz, 
which shows the function f is invariant under 7’, and has 
the same value at a pair of corresponding points of the plane. 
This function will vanish along the points of some conic K in 
the plane, which conic is invariant under all the collineations 
belonging to the group defined by R,. In order that this be 
a non-degenerate conic, we must assume D! # 0. 
The six relations R, on the nine elements of M leave three 
independent parameters for the group defined by R,. This is 
therefore a three-parameter group and will be designated 
by G,(K). 
THEOREM 14. The group defined by the quadratic relations R. 
is a three-parameter group and leaves invariant the conic whose 
equation is 
lx?-+ my?+ nz?+ 2pay + 2quz+ 2ryz = 0. 
211. Other Quadratic Relations. We saw in Art. 195 that 
the six equations of (24) imply another six, viz.: RP’, of (27). 
But since A = 17, the equations of R’, may be written thus: 
1A,2? ++ mB,\2+nC\2+ 2p AiBi+ 2q A1C, + 2r B,C, =1, 
1Ao?+ - = il; 
lA,?+ - - - - - =n, 
ps 3 
Ri’, * lA\Ao+ = = = = = =P, (27) 
lAyAs = : - - - =4, 
PARA gE 2 : : : - - 
