tinued at the
same time. But it is to be remarked that there is no ground for
believing, if this method of continuation be utilized, that the
function is single-valued; we may quite well return to the same values
of the independent variables with a different value of the function;
belonging, as we say, to a different branch of the function; and there
is even no reason for assuming that the number of branches is finite,
or that different branches have the same singular points and regions
of existence. Moreover, and this is the most difficult consideration
of all, all these circumstances may be dependent upon the values
supposed given to the arbitrary constants of the integral; in other
words, the singular points may be either _fixed_, being determined by
the differential equations themselves, or they may be _movable_ with
the variation of the arbitrary constants of integration. Such
difficulties arise even in establishing the reversion of an elliptic
integral, in solving the equation

/dx\^2
( -- ) = (x-a1)(x - a2)(x - a3)(x - a4);
\ds/

about an ordinary value the right side is developable; if we put x -
a1 = t1^2, the right side becomes developable about t1 = 0; if we put x
= 1/t, the right side of the changed equation is developable about t =
0; it is quite easy to show that the integral reducing to a definite
value x0 for a value s0 is obtainable by a series in integral powers;
this, however, must be supplemented by showing that for no value of s
does the value of x become entirely undetermined.

Linear differential equations with rational coefficients.

These remarks will show the place of the theory now to be sketched of
a particular class of ordinary linear homogeneous differential
equations whose importance arises from the completeness and generality
with which they can be discussed. We have seen that if in the
equations dy/dx = y1, dy1/dx = y2, ..., dy_n-2/dx = y_n-1,

dy_n-1/dx = a_n y + a_n-1 y1 + ... + a1 y_n-1,

where a1, a2, ..., an are now to be taken to be rational functions of
x, the value x = x^0 be one for which no one of these rational
functions is infinite, and y^0, y^01, ..., y^0_n-1 be quite arbitrary
finite values, then the equations are satisfied by

y = y^0u + y^01u1 + ... + y^0_n-1 u_n-1,

where u, u1, ..., un-1 are functions of x, independent of y^0, ...
y^0_n-1, developable about x = x^0; t