orresponding to any branch does not overlap itself, (2) that no two
such regions overlap, we have no space to enter. The second question
clearly requires the inquiry whether the group (that is, the monodromy
group) of the differential equation is properly discontinuous. (See
GROUPS, THEORY OF.)

The foregoing account will give an idea of the nature of the function
theories of differential equations; it appears essential not to exclude
some explanation of a theory intimately related both to such theories
and to transformation theories, which is a generalization of Galois's
theory of algebraic equations. We deal only with the application to
homogeneous linear differential equations.

Rationality group of a linear equation.

Irreducibility of a rational equation.

In general a function of variables x1, x2 ... is said to be rational
when it can be formed from them and the integers 1, 2, 3, ... by a
finite number of additions, subtractions, multiplications and
divisions. We generalize this definition. Assume that we have assigned
a fundamental series of quantities and functions of x, in which x
itself is included, such that all quantities formed by a finite number
of additions, subtractions, multiplications, divisions _and
differentiations in regard to x_, of the terms of this series, are
themselves members of this series. Then the quantities of this series,
and only these, are called _rational_. By a rational function of
quantities p, q, r, ... is meant a function formed from them and any
of the fundamental rational quantities by a finite number of the five
fundamental operations. Thus it is a function which would be called,
simply, rational if the fundamental series were widened by the
addition to it of the quantities p, q, r, ... and those derivable from
them by the five fundamental operations. A rational ordinary
differential equation, with x as independent and y as dependent
variable, is then one which equates to zero a rational function of y,
the order k of the differential equation being that of the highest
differential coefficient y^(k) which enters; only such equations are
here discussed. Such an equation P = 0 is called _irreducible_ when,
firstly, being arranged as an integral polynomial in y^(k), this
polynomial is not the product of other polynomials in y^(k) also of
rational form; and, secondly, the equation has no solution satisfying
also