30 
RADELEINGER. 
Eliminating the Y s by means of relation (1), we have 
— ( f 11 a n + ^12) 2 /i T" ( a i 2 ^11 T - ^12 ^22) 2/25 
^2 == (^21 ^11 H - g 21 6 22 ) 2 / 1 . + (^21 ^12 T - ^22 ^ 22 ) 2/21 
which is a substitution of the same form as the first. 
The general linear group can be written as follows : 
U ' *hi Vi "h a i 2 2/2 “h ^13 2/3 
^2 = «21 2/l + «22 2/2 + «23 2/3 
«ln 2/n 1 
«2n 2/n. 
= «nl 2/l + «n2 2/ 2 + <*n3 2/ 3 
«nn 2/n* J 
As long as a condition of the form (J) does not subsist between 
the ff s this is a group of n 2 independent parameters. 
The application of the theory of the general linear group 
to the linear differential equations is based on the idea of the 
irreducibility of these equations. This idea is also borrowed 
almost without change from the theory of algebraic equa¬ 
tions. Its extension to linear differential equations was first 
made by Frobenius.* The fundamental theorems demon¬ 
strated by him are as follows: 
(a) A linear differential equation is irreducible if it has no 
integral common with an equation of the same nature but of a 
lower degree. 
(b) When an equation is reducible there always exists an equa¬ 
tion of the same nature and of a lower degree which admits all of 
its integrals. 
(c) If an equation has one integral common with an irreduci¬ 
ble equation , it admits all of its integrals. 
Using these theorems as a basis, the two following funda¬ 
mental theorems were proven by Picard : 
(d) All rational f unctions of x and of y x ,y 2 , . y^and 
their derivatives which can be expressed as rational functions of x 
remain invariable when one performs on y Xi y„ . y n the 
substitutions of the general linear group. 
*Crelle’s Journal, vol. 76. 
