The formulae (1) and (15) might th u=1¢<»+£, §)(0.I)+3°:f'.%cf(0, I,2)-t 2°-$2 % $10, 1. 2. 3>+ oo,
u=u<>+£;§ '(O.I)+°5¥'~ % %(-1.0, I)+ x xo x x, x-x.,
T* -T Sh (-1, o, 1,2)-I-...(2o).
The general principle on which these formulae are constructed, and which may be used to construct other formulae, is that (1.) we start with any tabulated value of u, (11.) 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 tabular differences are used. If, for differences, we have
140-i-QL;-;9(o, 1) = ui
give the same result, provided
instance, we go only to first
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, t provided the differences are adjusted values of x for which u is tabulated be -yh, . . Then the table becomes
he above principle still applies, in a particular way. Let the
a=x0-i-ah, b=x0+Bh, c=x0+
if 7 K Adj usted Differences.” “D x. u.
1st Diff. 2nd Diff. &c.
i Q Q Q
a =x, , ua -b
(11, /9) (B)
=x u av 1 'Y
B B </an v>
c =-xy uy A
g .. ., I
ln this table, however, (a, B) does not mean ug -na, but (ug -ua) + (B-a); (G. 5, 1) means KB, v)~~(v1 /3)}+%(1f-11); and, generally
an uantit in the cblumn headed “rth diff." is yq y(n, ~—¢>>
obtained by dividing the difference the preceding column by (¢-17)/r. way, we may apply the principle of such as
u=1f, ,+°%9- (11, B)-I-33%
u=uy+%- (B. v)+T
of the adjoining quantities in
If the table is formed in this
§ 12 so as to obtain formulae
251. o, /1.~>+ cm,
<<», s, ~/>+... <22>.
The following example illustrates the method, h being taken to be 1°:-
Example 8. Y
Y - l 1st Diff. 2nd Diff. 3rd Dui. x' u"Sm x' (adjusted). (adjusted). (adjusted). l,
22° -3746066 I 1125 00,
161245 oo 7 48 75
23° -3907311 1 1222 50
158800 00 48 30
26° -4383711 1303 oo
G . 156194 00 47 49
27 4539905 1445 47
151857 60 46 oo
32° -5299195 1583 48 .
D 145523 67
35 '5735764 i 5 ww* i 4
To find it for x=3I°, we use the values for 26°, 27°, 32° and 35°, and obtain
u=-4383711 <>0+§ (156194 QOH? - i(-1445 47)+ Q if Ii(-46 OO)='51503801
which is only wrong in the last figure. en be written lf the values of u occurring in (21) or (22) are na, ug, u, /, . .u, , corresponding to values a, b, c, . . . l of x, the formula may be more symmetrically written
lf "b)( lf~'+<ic-ii)
ti-If) (yi-;...“» (23)-This
is known as Lagrangehv 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.-Nhen 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 A represents differencing by changing x into x-I-1, and A' differencing by changing y into 3/+I. Then the formula is Nzyy = (1 +A)”(1 +A')”u0,0;
and the right-hand side can be developed in whatever form is most convenient for the particular case.
REFERENCES.-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 rnath. Wiss. vol. i. pt. 2, pp. 800-819. For l. D. Everett's formula, see Qnar. four. 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 DIFFERENCES, CALCULUS or. (W. F. SH.)
INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek <j>pé1§ 'eL1/, to speak, or rrpdv-7-ew, 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 CONTRACT, STATUTE, WILL, &c.
'INTERREGNUM (Lat. inter, between, and regnurn, 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 ROME: History, ad init.). It was subsequently used in Republican times of an officer appointed to hold the cornitia 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 citriae. Under the Republic. similarly intcrreges 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 B.C., when the senate appointed an interrex to hold the cornitia which made