170 THEORY OF COLLINEATIONS. 
The group is a six-parameter group and will be designated 
by G, (1). 
207. Most General Linear Function of the Elements. Let 
us now assume the most general possible linear relation among 
the elements of M,, thus 
p = 1A, + mA,+nA3+ pBi + 98. +7rB8; +80, +tC, +ul;=v. 
Substitute for 4,, 8,, ete., their values from M, and set the co- 
efficients of «,,3,, etc., equal to the coefficients 4,, 8,, etc.; 
we thus obtain the following nine relations : 
la, + pbi+se=l, mat+tqryi+ti=m, na,+rb)+uca—n, 
laz+pbz+se.=p, ma2,+qb.+teo=q, nao +rbs+ue=r, (28) 
la; +pb;+sce3;=s, ma; + qb3+te,;=t , na,+rb;+uc3—=u. 
Our one assumed relation among the elements of M, leads to 
nine relations among the elements of M and evidently there is 
no group corresponding to our assumed relation. 
But ¢=v may be considered as the sum of three independ- 
ent relations as follows: 
1A, + p8,\+8s%=l, méAr.+q8.+tl.=q, nA3+7rB3;+nCl,=u. 
Each of these relations is in the form of a linear homogeneous 
function of the elements of a row of M, equated to a con- 
stant ; and we may therefore write down at once the three 
sets of relations, thus: 
1A,+ pB\+sCi\=1, mA,+q8,\+tCi\=m, nA,+7r8,+ul\=n, 
1A,+ pB.+sCs=p, mAr.+qB.+tCo=q, nA2+7rB.+uUli=r, ( 28’) 
lA, + pB. +sC3;=s, mA + q8;+ tC;=t, nA,+7rB8;+ul,—u. 
Applying to these three sets of relations the same process we 
applied to ¢=v we get the same nine relations as before and 
the correspondence is complete. We thus have the conditions 
for a group. 
These nine relations taken three at a time are equivalent to 
only six independent relations among eight independent par- 
ameters ; hence they define a two-parameter group G,. The 
geometric invariant of this group is readily seen by consider- 
ing the three relations down the first column of (28). These 
show that the point (/,p,s) is transformed into itself. In 
