dk2 + ... + kn φρk1 / dx2k2 ... dxnkn
reduces to bρk1 ... kn. Then the theorem is that there exists one, and only one, set of functions z1, ... zr, of x2, ... xn developable about a1, ... an satisfying the given differential equations, and such that for x1 = a1 we have
zσ = φσ, dzσ / dx1 = φσ(1), ... dhσ−1zσ / dhσ−1x1 = φσhσ−1.
And, moreover, if the arbitrary functions φσ, φσ(1) ... contain a certain number of arbitrary variables t1, ... tm, and be developable about the values tº1, ... tºm of these variables, the solutions z1, ... zr will contain t1, ... tm, and be developable about tº1, ... tºm.
The proof of this theorem may be given by showing that if ordinary power series in x1 − a1, ... xn − an, t1 − tº1, ... tm − tºm be substituted in the equations wherein in zσ the coefficients of (x1 − a1)º, x1 − a1, ..., (x1 − a1)hσ−1 are the arbitrary functions φσ, φσ(1), ..., φσh−1, divided respectively by 1, 1!, 2!, &c., then the differential equations determine uniquely all the other coefficients, and that the resulting series are convergent. We rely, in fact, upon the theory of monogenic analytical functions (see [Function]), a function being determined entirely by its development in the neighbourhood of one set of values of the independent variables, from which all its other values arise by continuation; it being of course understood that the coefficients in the differential equations are to be continued 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 Singular points of solutions. 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/ds)² = (x − a1)(x − a2)(x − a3)(x − a4);
about an ordinary value the right side is developable; if we put x − a1 = t1², 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.
These remarks will show the place of the theory now to be sketched of a particular class of ordinary linear homogeneous Linear differential equations with rational coefficients. 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, ..., dyn−2/dx = yn−1,
dyn−1/dx = any + an−1y1 + ... + a1yn−1,
where a1, a2, ..., an are now to be taken to be rational functions of x, the value x = xº be one for which no one of these rational functions is infinite, and yº, yº1, ..., yºn−1 be quite arbitrary finite values, then the equations are satisfied by