of the two solutions about x = 1, we should have found that x is not a single-valued function of ς′ unless λ2 is the inverse of an integer, or is zero; as ς′ is of the form (Aσ + B)/(Cς + D), A, B, C, D constants, the same is true in our case; equally, by considering the integrals about x = ∞ we find, as a third condition necessary in order that x may be a single-valued function of ς, that λ − μ must be the inverse of an integer or be zero. These three differences of the indices, namely, λ1, λ2, λ − μ, are the quantities which enter in the differential equation satisfied by x as a function of ς, which is easily found to be
| − | x111 | + | 3x²11 | ½(h − h1 − h2)x−1(x − 1)−1 + ½h1x−2 + ½h2(x − 1)−2, |
| x1³ | 2x14 |
where x1 = dx/dς, &c.; and h1 = 1 − y1², h2 = 1 − λ2², h3 = 1 − (λ − μ)². Into the converse question whether the three conditions are sufficient to ensure (1) that the σ region corresponding 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.
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 Rationality group of a linear equation. 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 Irreducibility of a rational equation. is not the product of other polynomials in y(k) also of rational form; and, secondly, the equation has no solution satisfying also a rational equation of lower order. From this it follows that if an irreducible equation P = 0 have one solution satisfying another rational equation Q = 0 of the same or higher order, then all the solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we can by differentiation express y(k+1), y(k+2), ... in terms of x, y, y(1), ... , y(k), and so put the function Q rationally in terms of these quantities only. It is sufficient, then, to prove the result when the equation Q = 0 is of the same order as P = 0. Let both the equations be arranged as integral polynomials in y(k); their algebraic eliminant in regard to y(k) must then vanish identically, for they are known to have one common solution not satisfying an equation of lower order; thus the equation P = 0 involves Q = 0 for all solutions of P = 0.
Now let y(n) = a1y(n−1) + ... + any be a given rational homogeneous linear differential equation; let y1, ... yn be n particular functions of x, unconnected by any equation with constant coefficients of the form c1y1 + ... + cnyn = 0, all satisfying The variant function for a linear equation. the differential equation; let η1, ... ηn be linear functions of y1, ... yn, say ηi = Ai1y1 + ... + Ainyn, where the constant coefficients Aij have a non-vanishing determinant; write (η) = A(y), these being the equations of a general linear homogeneous group whose transformations may be denoted by A, B, .... We desire to form a rational function φ(η), or say φ(A(y)), of η1, ... η, in which the η² constants Aij shall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the y1, ... yn were connected by a linear equation with constant coefficients. Such a function is in fact given, if the solutions y1, ... yn be developable in positive integral powers about x = a, by φ(η) = η1 + (x − a)n η2 + ... + (x − a)(n−1)n ηn. Such a function, V, we call a variant.
Then differentiating V in regard to x, and replacing ηi(n) by its value a1η(n−1) + ... + anη, we can arrange dV/dx, and similarly each of d²/dx² ... dNV/dxN, where N = n², as a linear function of the N quantities η1, ... ηn, ... η1(n−1), ... ηn(n−1), and The resolvent eqution. thence by elimination obtain a linear differential equation for V of order N with rational coefficients. This we denote by F = 0. Further, each of η1 ... ηn is expressible as a linear function of V, dV/dx, ... dN−1V / dxN−1, with rational coefficients not involving any of the n² coefficients Aij, since otherwise V would satisfy a linear equation of order less than N, which is impossible, as it involves (linearly) the n² arbitrary coefficients Aij, which would not enter into the coefficients of the supposed equation. In particular, y1 ,.. yn are expressible rationally as linear functions of ω, dω/dx, ... dN−1ω / dxN−1, where ω is the particular function φ(y). Any solution W of the equation F = 0 is derivable from functions ζ1, ... ζn, which are linear functions of y1, ... yn, just as V was derived from η1, ... ηn; but it does not follow that these functions ζi, ... ζn are obtained from y1, ... yn by a transformation of the linear group A, B, ... ; for it may happen that the determinant d(ζ1, ... ζn) / (dy1, ... yn) is zero. In that case ζ1, ... ζn may be called a singular set, and W a singular solution; it satisfies an equation of lower than the N-th order. But every solution V, W, ordinary or singular, of the equation F = 0, is expressible rationally in terms of ω, dω / dx, ... dN−1ω / dxN−1; we shall write, simply, V = r(ω). Consider now the rational irreducible equation of lowest order, not necessarily a linear equation, which is satisfied by ω; as y1, ... yn are particular functions, it may quite well be of order less than N; we call it the resolvent equation, suppose it of order p, and denote it by γ(v). Upon it the whole theory turns. In the first place, as γ(v) = 0 is satisfied by the solution ω of F = 0, all the solutions of γ(v) are solutions F = 0, and are therefore rationally expressible by ω; any one may then be denoted by r(ω). If this solution of F = 0 be not singular, it corresponds to a transformation A of the linear group (A, B, ...), effected upon y1, ... yn. The coefficients Aij of this transformation follow from the expressions before mentioned for η1 ... ηn in terms of V, dV/dx, d²V/dx², ... by substituting V = r(ω); thus they depend on the p arbitrary parameters which enter into the general expression for the integral of the equation γ(v) = 0. Without going into further details, it is then clear enough that the resolvent equation, being irreducible and such that any solution is expressible rationally, with p parameters, in terms of the solution ω, enables us to define a linear homogeneous group of transformations of y1 ... yn depending on p parameters; and every operation of this (continuous) group corresponds to a rational transformation of the solution of the resolvent equation. This is the group called the rationality group, or the group of transformations of the original homogeneous linear differential equation.
The group must not be confounded with a subgroup of itself, the monodromy group of the equation, often called simply the group of the equation, which is a set of transformations, not depending on arbitrary variable parameters, arising for one particular fundamental set of solutions of the linear equation (see [Groups, Theory of]).
The importance of the rationality group consists in three propositions. (1) Any rational function of y1, ... yn which is unaltered in value by the transformations of the group can be written in rational form. (2) If any rational function be changed The fundamental theorem in regard to the rationality group. in form, becoming a rational function of y1, ... yn, a transformation of the group applied to its new form will leave its value unaltered. (3) Any homogeneous linear transformation leaving unaltered the value of every rational function of y1, ... yn which has a rational value, belongs to the group. It follows from these that any group of linear homogeneous transformations having the properties (1) (2) is identical with the group in question. It is clear that with these properties the group must be of the greatest importance in attempting to discover what functions of x must be regarded as rational in order that the values of y1 ... yn may be expressed. And this is the problem of solving the equation from another point of view.
Literature.—(α) Formal or Transformation Theories for Equations of the First Order:—E. Goursat, Leçons sur l’intégration des équations aux dérivées partielles du premier ordre (Paris, 1891); E. v. Weber, Vorlesungen über das Pfaff’sche Problem und die Theorie der partiellen Differentialgleichungen erster Ordnung (Leipzig, 1900); S. Lie und G. Scheffers, Geometrie der Berührungstransformationen, Bd. i. (Leipzig, 1896); Forsyth, Theory of Differential Equations, Part i., Exact Equations and Pfaff’s Problem (Cambridge, 1890); S. Lie, “Allgemeine Untersuchungen über Differentialgleichungen, die eine continuirliche endliche Gruppe gestatten” (Memoir), Mathem. Annal.xxv. (1885), pp. 71-151; S. Lie und G. Scheffers, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (Leipzig, 1891). A very full bibliography is given in the book of E. v. Weber referred to; those here named are perhaps sufficiently representative of modern works. Of classical works may be named: Jacobi, Vorlesungen über Dynamik (von A. Clebsch, Berlin, 1866); Werke, Supplementband; G Monge, Application de l’analyse à la géométrie (par M. Liouville, Paris, 1850); J. L. Lagrange, Leçons sur le calcul des fonctions (Paris, 1806), and Théorie des fonctions analytiques (Paris, Prairial, an V); G. Boole, A Treatise on Differential Equations (London, 1859); and Supplementary Volume (London, 1865); Darboux, Leçons sur la théorie générale des surfaces, tt. i.-iv. (Paris, 1887-1896); S. Lie, Théorie der transformationsgruppen ii. (on Contact Transformations) (Leipzig, 1890).