1. The present paper is suggested by that of Dr. H. F. Baker in the ‘Proceedings of the London Mathematical Society,’ vol. xxxv., p. 333, “On the Integration of Linear Differential Equations.” In that paper a linear ordinary differential equation of order
n
is considered as derived from a system of
n
linear simultaneous differential equations
dx
i
/
dt
=
u
i1
x
+.....+
u
i
n
x
n
(
i
= 1...
n
), or, in abbreviated notation,
dx
/
dt
=
ux
, where
u
is a square matrix of
n
rows and columns whose elements are functions of
t
, and
x
denotes a column of
n
independent variables. A symbolic solution of this system is there given and denoted by the symbol Ω(
u
). This is a matrix of
n
rows and columns formed from
u
as follows :—Q (
ϕ
) is the matrix of which each element is the
t
-integral from
t
0
to
t
of the corresponding element of
ϕ
,
ϕ
being any matrix of
n
rows and columns; then Ω(
u
) = 1+Q
u
+Q
u
Q
u
+Q
u
Q
u
Q
u
.....
ad inf
., where the operator Q affects the whole of the part following it in any term.
Alternative integration procedure for scale-invariant ordinary differential equations
