LINEAR DIFFERENTIAL EQUATIONS. 31 
And the converse of this : 
(e) All rational functions of x and of y v y 2 , . y n and 
their derivatives which remain invariable by the substitutions of 
the general linear group are rational functions of x. 
As a necessary consequence of the last two propositions, it 
can be deduced that 
(f) The group of an irreducible equation of degree n contains, 
in general , n 2 independent parameters—that is, it coincides with 
the general linear group (K). 
Also: 
(g) If an equation of order n has less than n 2 independent 
parameters in its group, a rational relation subsists between its 
integrals y»y 2 , . y n and their derivatives, and the equa¬ 
tion is, in general, reducible. 
The considerations of a domain of rationality, so fruitful 
in the theory of algebraic equations, also apply. Hence re¬ 
sults a method of determining whether, in general, the inte¬ 
grals of one equation are expressible rationally, in terms, of 
those of another equation. 
The application of the theory founded on the foregoing 
theorems furnish us with no means of solving new equa¬ 
tions—that is, no means of actually calculating their inte¬ 
grals. It does, however, establish a foundation on which we 
can base a classification of the linear differential equations. 
This classification can be accomplished by determining all 
the subgroups of the general linear groups corresponding to 
the equations of orders from 2 to n and the corresponding 
reduced equations. This method is tedious and has thus 
far been completely carried out for orders 2 and 3 only, but 
the results are sure, and many, and important ones may be 
expected from its future extension. 
The following is an outline of some of the results obtained 
by applying the method to the simplest case, n = 2. The 
general linear group corresponding to this case has four 
independent parameters. 
It is: 
hi = On Vi 4- ■ 
^2 = «21 Vi + «22 2/2* 
