ANALYTIC CONDITIONS. 151 
where l,’, l,’, ete., are functions only of l,, 1,, ete., and the 
elements of M and may be written 9;(a,,b,,.. 31,1, ..)=l,. 
This gives us a set of relations among the elements of M. 
But this set of relations thus obtained among the elements of 
M must be, at least, a part of the set of assumed relations 
Te (Gp 0) 1. i, Gs 9») 1G.) We cannot. in this'way get 
more relations on the elements of M than our assumed set, 
for by assumption all the relations among’the elements of M 
are also satisfied by the elements of M, and M,. We may 
get by this process exactly the assumed set of relations or a 
part of them. Evidently there is one and only one function 
¢ for each parameter / in the set R; and among the constants 
c, are the 7 parameters /;. The parts played by M and M, 
may be interchanged in the above process. 
Let us select one equation ¢,=1, of the set and investigate 
the parameters contained in the function ~,. The function 
fa May contain all of the 7 parameters or only a part of them. 
First let us suppose that }, contains all of the 7 parameters. 
Then transforming }, by U and equating parameters as above 
we get the whole set of equations ~;=1;. The function ¢, re- 
peats itself in /,’ which is therefore by this process a function 
of all the parameters including /,; presumably the rest of 
the parameters of the transformed function are also functions 
of all the parameters including /,. Any one of these trans- 
formed parameters as 1,’ that contains /, contains all the pa- 
rameters ; for transforming »,=1, by U we get again 1,’=,. 
But by this process 1,’ is a function only of the parameters in 
g, and the elements of M. Since we know that l, contains all 
the parameters, @, must contain all of them. In like manner 
every function ¢,, 4, ... in the set that contains l, con- 
tains all the parameters, and from each of these functions can 
be generated all the functions of the set ¢;. The set of equa- 
tions p;= 1; is therefore a self-generating set and no equation 
not already in the set can be generated from any equation of 
the set. The system of collineations defined by 9,=1, there- 
fore has the first group property. 
