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Rh The formulae (1) and (15) might then be written

(19),

(20).

The general principle on which these formulae are constructed, and which may be used to construct other formulae, is that (i.) we start with any tabulated value of u, (ii.) we pass to the successive differences by steps, each of which may be either downwards or upwards, and (iii.) the new suffix which is introduced at each step determines the new factor (involving x) for use in the next term. For any particular value of x, however, all formulae which end with the same difference of the rth order give the same result, provided tabular differences are used. If, for instance, we go only to first differences, we have

identically.

13. Ordinates not Equidistant.—When the successive ordinates in the graph of u are not equidistant, i.e. when the differences of successive values of x are not equal, the above principle still applies, provided the differences are adjusted in a particular way. Let the values of x for which u is tabulated be a = x0 + h, b = x0 + h, c = x0 + h, Then the table becomes

In this table, however, does not mean uundefined − uundefined, but uundefined − uundefined ÷ ( − ); means { − } ÷ ( − ); and, generally any quantity in the column headed “rth diff.” is obtained by dividing the difference of the adjoining quantities in the preceding column by ( − )/r. If the table is formed in this way, we may apply the principle of § 12 so as to obtain formulae such as

(21),

(22).

The following example illustrates the method, h being taken to be 1°:—

Example 8.

To find u for x = 31°, we use the values for 26°, 27°, 32° and 35°, and obtain

which is only wrong in the last figure.

If the values of u occurring in (21) or (22) are uundefined, uundefined, uundefined, uundefined, corresponding to values a, b, c, l of x, the formula may be more symmetrically written

(23).

This is known as Lagrange’s formula, but it is said to be due to Euler. It is not convenient for practical use, since it does not show how many terms have to be taken in any particular case.

14. Interpolation from Tables of Double Entry.—When u is a function of x and y, and is tabulated in terms of x and of y jointly, its calculation for a pair of values not given in the table may be effected either directly or by first forming a table of values of u in terms of y for the particular value of x and then determining u from this table for the particular value of y. For direct interpolation, consider that represents differencing by changing x into x + 1, and ′ differencing by changing y into y + 1. Then the formula is

ux, y = (1 + )x (1 + ′)y u0,0;

and the right-hand side can be developed in whatever form is most convenient for the particular case.

—For general formulae, with particular applications, see the Text-book of the Institute of Actuaries, part ii. (1st ed. 1887, 2nd ed. 1902), p. 434; H. L. Rice, Theory and Practice of Interpolation (1899). Some historical references are given by C. W. Merrifield, “On Quadratures and Interpolation,” ''Brit. Assoc.'' Report (1880), p. 321; see also ''Encycl. der math. Wiss.'' vol. i. pt. 2, pp. 800-819. For J. D. Everett’s formula, see ''Quar. Jour. Pure'' and Applied Maths., No. 128 (1901), and ''Jour. Inst. Actuaries'', vol. xxxv. (1901), p. 452. As to relative accuracy of different formulae, see ''Proc. Lon. Math. Soc.'' (2) vol. iv. p. 320. Examples of interpolation by means of auxiliary curves will be found in ''Jour. Royal Stat. Soc.'' vol. lxiii. pp. 433, 637. See also.

INTERPRETATION (from Lat. interpretari, to expound, explain, interpres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek , to speak, or  , to do), in general, the action of explaining, or rendering the sense of an obscure form of words or an unknown tongue into a language comprehended by the person addressed. In legal use the word “interpretation” is employed in the sense of ascertaining the meaning of the language of a document, as well as its relation to facts. It is also applied to acts of parliament, as pointing out the sense in which particular words used therein are to be understood. The interpretation of documents and statutes is subject to definite legal rules, the more important of which will be found in the articles, , , &c.

INTERREGNUM (Lat. inter, between, and regnum, reign), strictly a period during which the normal constituted authority is in abeyance, and government is carried on by a temporary authority specially appointed. Though originally and specifically confined to the sphere of sovereign authority, the term is commonly used by analogy in other connexions for any suspension of authority, during which affairs are carried on by specially appointed persons. The term originated in Rome during the regal period when an interrex was appointed (traditionally by the senate) to carry on the government between the death of one king and the election of his successor (see : History, ad init.). It was subsequently used in Republican times of an officer appointed to hold the comitia for the election of the consuls when for some reason the retiring consuls had not done so. In the regal period when the senate, instead of appointing a king, decided to appoint interreges, it divided itself into ten decuries from each of which one senator was selected. Each of these ten acted as king for five days, and if, at the end of fifty days, no king had been elected, the rotation was renewed. It was their duty to nominate a king, whose appointment was then ratified or refused by the curiae. Under the Republic similarly interreges acted for five days each. When the first consuls were elected (according to Dionysius iv. 84 and Livy i. 60), Spurius Lucretius held the comitia as interrex, and from that time down to the Second Punic War such officers were from time to time appointed. Thenceforward there is no record of the office till 82, when the senate appointed an interrex to hold the comitia which made