150 THEORY OF COLLINEATIONS. 
In like manner we see that the substitutions for the ele- 
ments in the columns of M. are in. the form of linear trans- 
formations in the elements of the corresponding columns of 
M, the matrix of the transformation being M,. The trans- 
formation on the first column is written thus : 
A, =a,4,+ /1d2+714s , 
V, 5 Ay = 420, + 202+ 724s , (19) 
As = 420) + 3302 +73As 5 
and the others similarly. Hence the following theorem : 
THEOREM 7. The substitutions for the elements of M2 in terms 
of those of Mand M, are, in one of two ways, in the form of the 
same linear transformation in three sets of three variables each. 
The three sets of cogredient variables are the elements of the three 
rows of M, (or columns of M) while the matrix of the transforma- 
tion is the conjugate of M (or M, itself). 
184. The Set of Relations R. Our problem is now to de- 
termine the properties of a set of relations on the elements of 
M such that their existence is both a necessary and a sufficient 
condition for a subgroup of G,. Let us assume the existence 
of a system of collineations within G, having the first group 
property and hence a set of relations that are all satisfied by 
the elements of M, M,, and M,. Let the set R be y in num- 
ber and let them be represented by F’,(@,,0,, .- 34, . -)=Crs 
(k=1,2,..7), where l,, l. .. l,are constants or parameters 
of the functions F’,. It is conceivable that a given function 
F of the set may contain all, a part, or none of the parame- 
ters of the set; also that one or more of the F’s may contain 
all, a part, or none of the elements of M. Let us substitute 
for 4,, B,, etc., in these relations on M, their values from equa- 
tion (10) in terms of the elements of Mand M,. Consider 
the new relations thus formed as functions of «,, 3,, etc., 
the elements of M,. Since these new relations on «,, /3,, ete., 
are by hypothesis of exactly the same form as the original 
relations on 4,, 8,, etc., they may be written F',(a,, 3,,... ; 
L/, l,/,...1/) =e, and we can equate the corresponding pa- 
rameters of the two sets. We thus get l,/=1,, l,’=l., ete., 
