444 
[862 
862. 
NOTE ON THE THEORY OF LINEAR DIFFERENTIAL EQUATIONS. 
[From Crelles Journal dev Mathem., t. c. (1887), pp. 286—295.] 
1. I CONSIDER a linear differential equation 
d m y d m ~ 1 y 
P°d5i>+P'dii^ + - + P™y = 0 ’ 
where p 0 , pi, • •• > Pm are rational and integral functions of x, having no common factor: 
as usual, x is a complex magnitude represented by a point, and we consider the 
integrals belonging to a singular point x=a of the differential equation. An integral 
may be a regular integral, or it may be what Thome calls a normal elementary 
integral: the theory of these integrals (which I would rather call subregular integrals) 
requires, I think, further examination. 
2. I retain x as the independent variable, but for shortness use t to denote 
—-— : and I take as the dependent variable z, = - ~ : we have then z determined 
x — a y dx 
by a non-linear differential equation of the order m — 1, and from any value of z we 
derive a corresponding value e$ zdx of y. 
3. To obtain the ^-equation, using for shortness accents to denote differentiation 
in regard to x, we have z = ' 
hence 
also 2 zz' + z" = ~ , that is, 
y y 
0 (z 2 + z) + 2zz' + z" = — , or finally z 3 + 3zz' + z" = —, 
y y 
y y y 
and so on ; viz. the values of J 
y y y 
are z, z 2 + z', z 3 + 3zz' + z", 
generally for