is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we can express y and z in terms of u, v, and x (the attempt to do this could only fail by leading to a relation connecting u, v and x, and the existence of such a relation would involve that the determinant
| ∂u | ∂v | − | ∂u | ∂v |
| ∂y | ∂z | ∂z | ∂y |
was zero), and so write F in the form F(x, y, z) = Φ(u, v, x). We then have
| ∂F | = | ∂Φ | ∂u | + | ∂Φ | ∂v | + | ∂Φ | , | ∂F | = | ∂Φ | ∂u | + | ∂Φ | ∂v | , | ∂F | = | ∂Φ | ∂u | + | ∂Φ | ∂v | ; |
| ∂x | ∂u | ∂x | ∂v | ∂x | ∂x | ∂y | ∂u | ∂y | ∂v | ∂y | ∂z | ∂u | ∂z | ∂v | ∂z |
thereby the Jacobian determinant of F, u, v is reduced to
| ∂Φ | ( | ∂u | ∂v | − | ∂u | ∂v | ); |
| ∂x | ∂y | ∂z | ∂z | ∂y |
by hypothesis the second factor of this does not vanish identically; hence ∂Φ/∂x = 0 identically, and Φ does not contain x; so that F is expressible in terms of u, v only; as was to be proved.
Part II.—General Theory.
Differential equations arise in the expression of the relations between quantities by the elimination of details, either unknown or regarded as unessential to the formulation of the relations in question. They give rise, therefore, to the two closely connected problems of determining what arrangement of details is consistent with them, and of developing, apart from these details, the general properties expressed by them. Very roughly, two methods of study can be distinguished, with the names Transformation-theories, Function-theories; the former is concerned with the reduction of the algebraical relations to the fewest and simplest forms, eventually with the hope of obtaining explicit expressions of the dependent variables in terms of the independent variables; the latter is concerned with the determination of the general descriptive relations among the quantities which are involved by the differential equations, with as little use of algebraical calculations as may be possible. Under the former heading we may, with the assumption of a few theorems belonging to the latter, arrange the theory of partial differential equations and Pfaff’s problem, with their geometrical interpretations, as at present developed, and the applications of Lie’s theory of transformation-groups to partial and to ordinary equations; under the latter, the study of linear differential equations in the manner initiated by Riemann, the applications of discontinuous groups, the theory of the singularities of integrals, and the study of potential equations with existence-theorems arising therefrom. In order to be clear we shall enter into some detail in regard to partial differential equations of the first order, both those which are linear in any number of variables and those not linear in two independent variables, and also in regard to the function-theory of linear differential equations of the second order. Space renders impossible anything further than the briefest account of many other matters; in particular, the theories of partial equations of higher than the first order, the function-theory of the singularities of ordinary equations not linear and the applications to differential geometry, are taken account of only in the bibliography. It is believed that on the whole the article will be more useful to the reader than if explanations of method had been further curtailed to include more facts.
When we speak of a function without qualification, it is to be understood that in the immediate neighbourhood of a particular set x0, y0, ... of values of the independent variables x, y, ... of the function, at whatever point of the range of values for x, y, ... under consideration x0, y0, ... may be chosen, the function can be expressed as a series of positive integral powers of the differences x − x0, y − y0, ..., convergent when these are sufficiently small (see [Function: Functions of Complex Variables]). Without this condition, which we express by saying that the function is developable about x0, y0, ..., many results provisionally stated in the transformation theories would be unmeaning or incorrect. If, then, we have a set of k functions, ƒ1 ... ƒk of n independent variables x1 ... xn, we say that they are independent when n ≥ k and not every determinant of k rows and columns vanishes of the matrix of k rows and n columns whose r-th row has the constituents dƒr/dx1, ... dƒr/dxn; the justification being in the theorem, which we assume, that if the determinant involving, for instance, the first k columns be not zero for x1 = xº1 ... xn = xºn, and the functions be developable about this point, then from the equations ƒ1 = c1, ... ƒk = ck we can express x1, ... xk by convergent power series in the differences xk+1 − xºk+1, ... xn − xnº, and so regard x1, ... xk as functions of the remaining variables. This we often express by saying that the equations ƒ1 = c1, ... ƒk = ck can be solved for x1, ... xk. The explanation is given as a type of explanation often understood in what follows.