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\chapter{On Partial Differential Equations}

\section{Preliminary Notice On Integration}

In all that follows, we shall suppose that it is always possible to effect simple integration; inasmuch as any difficulties of integration, connected with the solutions of partial differential equations, do not affect the principle of those solutions. Thus, for instance, we shall not hesitate to represent an unknown function of $x$ by $\chi''(x)$, (the second differential-coefficient of $\chi(x)$), on the assumption that, whatever be the form of $\chi''(x)$, we can in some way find the function $\chi(x)$ of which it is the second differential-coefficient.

\section{Characteristics Of The Solutions Of Simple Differential Equations}

Before entering on the subject of Partial Differential Equations, it may be convenient to consider some of the characteristics of the solutions of Simple Differential Equations.

To begin with Simple Differential Equations of the first order. Suppose, for facility of geometrical illustration, we consider the equation $y\frac{dy}{dx}=a$, or the equation $y\frac{dx}{dy}=b$, of which the former, translated geometrically, indicates that the subnormal of a plane-curve (to be found) is constant, and the latter indicates that the subtangent is constant. The algebraical solutions are easily found: in each, there is a constant, and is not defined by it; a constant of that class described (perhaps improperly) by the term ``arbitrary,'' but which really means ``not yet determined, but enabling us by  proper determination of its value so to fix the value of $x$ corresponding to a given value of $y$ that we can adjust the solution to some specific condition.''

The reader is requested to observe that instead of the term ``arbitrary constant,'' we shall always use the term ``undetermined constant.''

If we treat the geometrical translation of the differential equation by a geometrical translation of the differential equation by a geometrical process, always drawing a normal or tangent so as to make the subnormal or subtangent constant, then drawing by means of it a small portion of the curve, then repeating the process, \&c., we may produce a polygon which will approach to the strict solution, with smaller errors (by taking the sides small enough) than any small quantity that can be assigned. In each case, however, a starting point is necessary.

In both ways of treating the problems these conditions manifestly hold:

  \item The curve, in each case, is one definite curve.
  \item The curve, in each case, is a continuous curve, expressed by the same equation through its whole extent.
  \item Even if there be isolated points or curves, still the same one equation defines the whole.
  \item It is necessary to introduce on undetermined quantity, enabling us to adjust the curve to a specific condition given by special considerations: that undetermined quantity is however a simple constant.

Let us now consider a Simple Differential Equation of the second order: such for instance as is given by this problem, ``To find the curve in which the radius of curvature is a function, given in form, of the ordinate.'' here the algebraic solution gives a formula requiring two undetermined quantities, still simple constants: the equivalent geometrical treatment shews that we require two elements, (as for instance, the value of $x$ and the inclination of the tangent, for some one value of $y$). But all the conditions hold which are mentioned in Article 5: the only difference being that, instead of \emph{one} undetermined quantity, a simple constant, there must now be \emph{two} undetermined quantities, simple constants.

The conditions which will be found to hold in the solutions of Partial Differential Equations differy very remarkably from these.