Henry Frederick Baker

“ 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. ”

“ When λ1 is a positive integer, not zero, the second solution about x = 0 persists as a solution, in accordance with the order of arrangement of the roots of the index equation in our theory; the first solution is then replaced by an integral polynomial of degree -λ or −μ1, when λ or μ is one of the negative integers 0, −1, −2, ..., 1 − λ1, but otherwise contains a logarithm. ”

“ Let these n − 1 functions together with x1 be introduced as n new independent variables in all the r equations. Since the first equation is satisfied by n − 1 of the new independent variables, it will contain no differential coefficients in regard to them, and will reduce therefore simply to dƒ/dx1 = 0, expressing that any common solution of the r equations is a function only of the n − 1 remaining variables. Thereby the investigation of the common solutions is reduced to the same problem for r − 1 equations in n − 1 variables. ”

“ From this equation and the n − 1 equations Πiω = 0, Pσω = 0, we can determine ω by a quadrature only. Hence can be deduced a much more general result, that if the group of r parameters be integrable, the complete system can be entirety solved by quadratures; it is only necessary to introduce the solution found by the first quadrature as an independent variable, whereby we obtain a complete system of n − r equations in n − 1 variables, subject to an integrable group of r − 1 parameters, and to continue this process. ”

“ Assuming the condition in its general form, take in the given differential equation a plane section of the surface φ = C parallel to the plane z, viz. put z constant, and consider the resulting differential equation in the two variables x, y, namely Xdx + Ydy = 0; let ψ (x, y, z) = constant, be its integral, the constant z entering, as a rule, in ψ because it enters in X and Y. Now differentiate the relation ψ (x, y, z) = ƒ (z) , where ƒ is a function to be determined, so obtaining
there exists a function σ of x, y, z such that
because ψ = constant, is the integral of Xdx + Ydy = 0 ”

“ Passing now to the case of three variables, suppose that the Jacobian determinant of the three functions F, u, v in regard to x, y, z is identically zero. We prove that if u, v are not themselves functionally connected, F is expressible as a function of u and v. Suppose first that the minors of the elements of ∂F/∂x, ∂F/∂y, ∂F/∂z in the determinant are all identically zero, namely the three determinants such as
then by the case of two variables considered above there exist three functional relations. ”

“ 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 ”

“ The index equation at x = ξ1 is r (r − 1) + rψ (ξ1) / φ′ (ξ1) + θ (ξ) 1 / [φ′ (ξ1) ] 2 = 0, and if α1, β1 be its roots, we have α1 + β1 = 1 − ψ (ξ1) / φ′ (ξ1) and α1β1 = θ (ξ) 1 / [φ′ (ξ1) ] 2. Thus by an elementary theorem of algebra, the sum Σ (1 − αi − βi) / (x − ξi) , extended to the m finite singular points, is equal to ψ (x) / φ (x) , and the sum Σ (1 − αi − βi) is equal to the ratio of the coefficients of the highest powers of x in ψ (x) and φ (x) , and therefore equal to 1 + α + β, where α, β are the indices at x = ∞. ”

“ When in a given set of differential equations the number of equations is greater than the number of dependent variables, the equations cannot be expected to have common solutions unless certain conditions of compatibility, obtainable by equating different forms of the same differential coefficients deducible from the equations, are satisfied. ”

“ Z − z=p (X − x) + q (Y − y)
passing through it, where X, Y, Z are current co-ordinates, and call this association a surface-element; and if two consecutive elements of which the point (x + dx, y + dy, z + dz) of one lies on the plane of the other, for which, that is, the condition dz = pdx + qdy is satisfied, be said to be connected, and an infinity of connected elements following one another continuously be called a connectivity, then our statement is that a connectivity consists of not more than ∞2 elements, the whole number of elements (x, y, z, p, q) that are possible being called ∞5. ”

“ Now consider a region enclosing xo and only one of the places, say Σ, at which one of a1, . . . an becomes infinite. When x is made to describe a closed curve in this region, including this point Σ in its interior, it may well happen that the continuations of the functions u, u1, . . ., un−1 give, when we have returned to the point x, values v, v1, . . ., vn−1, so that the integral under consideration becomes changed to yo + yo1v1 + . ”

“ It is seen at once that this result is a generalization of the theorem for r = 1, and its proof is conveniently given by induction from that case. It can be verified without difficulty (1) that if from the r equations of the complete system we form r independent linear aggregates, with coefficients not necessarily constants, the new system is also a complete system; (2) that if in place of the independent variables x1, ... xn we introduce any other variables which are independent functions of the former, the new equations also form a complete system. ”

“ Putting each of the fractions in the characteristic equations equal to dt, the equations enable us, starting from an arbitrary element x′0, y′0, z′0, p′0, q′0, about which all the quantities F′, dF′/dp′, &c., occurring in the denominators, are developable, to define, from the differential equation F′=0 alone, a connectivity of ∞1 elements, which we call a characteristic chain ”

“ 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 ”

“ The most natural method of obtaining a suggestion lies probably in remarking that if r2 = r1 + h, there is an integral [ (x − ξ) r1 + hφ2 − (x − ξ) r1φ1] / h, where the coefficients in φ2 are the same functions of r1 + h as are the coefficients in φ1 of r1; when h vanishes, this integral takes the form
(x − ξ) r1 [dφ1/dr1 + φ1 log (x − ξ) ] ,
or say
(x − ξ) r1 [φ1 + ψ1 log (x − ξ) ] ;
denoting this by 2πiμ1K, and (x − ξ) r1 φ1 by H, a circuit of the point ξ changes K into
A similar artifice suggests itself when three of the roots of the determinantal equation are the same, and so on. ”

“ There is clearly such a set for each of the values s = 0, s = 1, . . ., s = m−1, s = m. And for any value of s there may exist relations additional to the specified m + 1 relations, provided they do not involve any relation connecting t, t1, . . . tm only, and are consistent with the m−s relations connecting u1, ... um. It is now evident that, essentially, the integration of a Pfaffian equation
a1dx1 + ... + andxn=0,
wherein a1, ... an are functions of x1, ... xn, is effected by the processes necessary to bring it to its reduced form, involving only independent variables. ”

“ In the general case the equations
dxj=c1j dx1 + . .crjdxr
similarly determine through an arbitrary point x1o, . . . xno Mayer’s method
of integration. a planar manifold of r dimensions in space of n dimensions, and when the conditions of integrability are satisfied, every direction in this manifold through this point is tangent to the manifold of r dimensions, expressed by ωr+1 = x0r+1, . . . ωn = xno, which satisfies the equations and passes through this point. ”

“ In particular, y1,.. yn are expressible rationally as linear functions of ω, dω/dx, . . . d N−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. ”

“ 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) , . . . ”

“ Consider the matrix of four rows and three columns, in which the elements of the first row are a, b, c, and the elements of the (r + 1) -th row, for r = 1, 2, 3, are the quantities ur1, ur2, ur3, where u11 = u22 = u33 = 0. Then it is easily seen that the cases (1) , (2) , (3) above correspond respectively to the cases when (1) every determinant of this matrix of two rows and columns is zero, (2) every determinant of three rows and columns is zero, (3) when no condition is assumed. This result can be generalized as follows: if a1, ... an be any functions of x1, ... ”

“ 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) ”