ANALYTIC CONDITIONS. 149 
The existence of subgroups of G, is evident geometrically. 
It was shown in Theorem 1 of this chapter that the invariance 
of a geometric figure under all the transformations of a cer- 
tain system is a sufficient condition that they form a group. 
This may be expressed analytically by saying that the inva- 
riance under T of one or more functions of the variables is a 
sufficient condition for a subgroup of G,. If a certain func- 
tion of the variables remains invariant under a linear trans- 
formation T, then the corresponding parameters of the 
original and of the transformed functions must be equal. 
This implies that one or more relations exist among the ele- 
ments of the matrix M, and it also gives us a hint as to the 
form of such relations. Thus it appears that the existence of 
a proper set of relations among the elements of M is a suffi- 
cient condition for a subgroup of G,. 
183. The Substitutions U and V. We must first ex- 
amine the form assumed by the substitution for the elements 
of M, in terms of those of Mand M,. We note first the form 
assumed by the substitution for the elements of the first row 
of M,. From the two forms of M,, equation (10), we have 
the substitution, 
A, =a, 4, + 42/7; +43" , 
U, : Si =biatbofitbsn , (18) 
Ci = C14, + 62/3; +3") ; 
which is in the form of a linear transformation, the variables 
being the elements of the first row of M,, and the matrix of 
the transformation being the conjugate of M@. We also note 
that the substitutions for the second and third rows of M, 
give us the same linear transformation so far as the matrix 
is concerned, the variables being the elements of the second 
and third rows of M, respectively. The elements of the three 
rows of M, form therefore three sets of cogredient variables. 
The three substitutions may be expressed by one formula, 
thus : 
A, =4,4;4+ a2/3,+ 437i ’ 
OF 5 B; = 61%; + b2/5;+ b37i , (i=1,2,3) (18’) 
Ci = 14,4 Co fi tesr: ; 
