Page:EB1911 - Volume 14.djvu/737

Rh common law courts in interpleader; and the Judicature Act 1875 enacted that the practice and procedure under these two statutes should apply to all divisions of the High Court of Justice. The Judicature Act also extended the remedy of interpleader to a debtor or other person liable in respect of a debt alleged to be assigned, when the assignment was disputed. In 1883 the acts of 1831 and 1860 were embodied in the form of rules by the Rules of the Supreme Courts (1883), O. lvii. by reference to which all questions of interpleader in the High Court of Justice are now determined. The acts themselves were repealed by the Statute Law Revision Act of the same year. Interpleader is the equivalent of multiplepoinding in Scots law.

INTERPOLATION (from Lat. interpolare, to alter, or insert something fresh, connected with polire, a polish), in mathematics, the process of obtaining intermediate terms of a series of which particular terms only are given. The cubes, for instance, shown in the second column of the accompanying table, may be regarded as terms of a series, and the cube of a fractional number, not exceeding the last number in the first column, may be found by interpolation. The process of obtaining the cube of a number exceeding the last number in the first column would be extrapolation; the formulae which apply to interpolation apply in theory to extrapolation, but in practice special precautions as to accuracy are necessary. The present article deals only with interpolation.

The term is usually limited to those cases in which there are two quantities, x and u, which are so related that when x has any arbitrary value, lying perhaps between certain limits, the value of u is determinate. There is a given series of associated values of u and of x, and interpolation consists in determining the value of u for any arbitrary value of x, or the value of x for any arbitrary value of u, lying between two of the values in the series. Either of the two quantities may be regarded as a function of the other; it is convenient to treat one, x, as the “independent variable,” the other, u, being treated as the “dependent variable,” i.e. as a function of x. If, as is usually the case, the successive values of one of the quantities proceed by a constant increment, this quantity is to be regarded as the independent variable. The two series of values may be tabulated, those of x being placed in a column (or row), and those of u in a parallel column (or row); u is then said to be tabulated in terms of x. The independent variable x is called the argument, and the dependent variable u is called the entry. Interpolation, in the ordinary sense, consists in determining the value of u for a value of x intermediate between two values appearing in the table. This may be described as direct interpolation, to distinguish it from inverse interpolation, which consists in determining the value of x for a value of u intermediate between two in the table. The methods employed can be extended to cases in which the value of u depends on the values of two or more independent quantities x, y,...

In the ordinary case we may regard the values of x as measured along a straight line OX from a fixed point O, so that to any value of x there corresponds a point on the line. If we represent the corresponding value of u by an ordinate drawn from the line, the extremities of all such ordinates will lie on a curve which will be the graph of u with regard to x. Interpolation therefore consists in determining the length of the ordinate of a curve occupying a particular position, when the lengths of ordinates occupying certain specified positions are known. If u is a function of two variables, x and y, we may similarly represent it by the ordinate of a surface, the position of the ordinate being determined by the values of x and of y jointly.

The series or tables to which interpolation has to be applied may for convenience be regarded as falling into two main groups. The first group comprises mathematical tables, i.e. tables of mathematical functions; in the case of such a table the value of the function u for each tabulated value of x is calculated to a known degree of accuracy, and the degree of accuracy of an interpolated value of u can be estimated. The second group comprises tables of values which are found experimentally, e.g. values of a physical quantity or of a statistical ratio; these values are usually subject to certain “errors” of observation or of random selection (see ). The methods of interpolation are usually the same in the two groups of cases, but special considerations have to be taken into account in the second group. The line of demarcation of the two groups is not absolutely fixed; the tables used by actuaries, for instance, which are of great importance in practical life, are based on statistical observations, but the tables formed directly from the observations have been “smoothed” so as to obtain series which correspond in form to the series of values of mathematical functions.

It must be assumed, at any rate in the case of a mathematical function, that the “entry” u varies continuously with the “argument” x, i.e. that there are no sudden breaks, changes of direction, &c., in the curve which is the graph of u.

Various methods of interpolation are described below. The simplest is that which uses the principle of proportional parts; and mathematical tables are usually arranged so as to enable this method to be employed. Where this is not possible, the methods are based either on the use of Taylor’s Theorem, which gives a formula involving differential coefficients (see ), or on the properties of finite differences (see ). Taylor’s Theorem can only be applied directly to a known mathematical function; but it can be applied indirectly, by means of finite differences, in various cases where the form of the function expressing u in terms of x is unknown; and even where the form of this function is known it is sometimes more convenient to determine the differential coefficients by means of the differences than to calculate them directly from their mathematical expressions. Finally, there are cases where we cannot even employ finite-difference formulae directly. In these cases we must adopt some special method; e.g. we may instead of u tabulate some function of u, such as its logarithm, which is found to be amenable to ordinary processes, then determine the value of this function corresponding to the particular value of x, and thence determine the corresponding value of u itself.

In considering methods of interpolation, it will be assumed, unless the contrary is stated, that the values of x proceed by a constant increment, which will be denoted by h.

In order to see what method is to be employed, it is usually necessary to arrange the given series of values of u in the form of a table, as explained above, and then to take the successive differences of u. The differences of the successive values of u are called its first differences; these form a new series, the first differences of which are the second differences of u; and so on. The systems of notation of the differences are explained briefly below. For the fuller discussion, reference should be made to.

A. Direct Interpolation.

1. Interpolation by First Differences.—The simplest cases are those in which the first difference in u is constant, or nearly so. For example:—

In Example 1 the first difference of u corresponding to a difference of h ≡ .001 in x is .0001000; but, since we are working throughout to seven places of decimals, it is more convenient to write it 1000. This system of ignoring the decimal point in dealing with differences will be adopted throughout this article. To find u for an intermediate value of x we assume the principle of proportional parts, i.e. we assume that the difference in u is proportional to the difference in x. Thus for x = 4.342945 the difference in u is .945 of 1000 = 945, so that u is .6376898 + .0000945 = .6377843. For x = 4.34294482 the difference in u would be 944.82, so that the value of u would apparently be .6376898 + .000094482 = .637784282. This, however, would be incorrect. It must be remembered that the values of u are only given “correct to seven places of decimals,” i.e. each