tabulated value differs from the corresponding true value by a tabular error which may have any value up toié of -ooooool; and we
cannot therefore by interpolation obtain a result which is correct
to nine places. If the interpolated value of 14 has to be used in
calculations for which it is important that this value should be as
accurate as possible, it may be convenient to retain it temporarily
in the form ~6 '§ 76898+944 82=-6377842 82 or -6376898-{~944“2=
-637784235 but we must ultimately return to the seven-place
arrangement and write it as -6377843. The result of interpolation
by first difference is thus usually subject to two inaccuracies, the
first being the tabular error of 14 itself, and the second being due to
the necessity of adjusting the final figure of the added (proportional)
difference. If the tabulated values are correct to seven places of
decimals, the interpolated value, with the final figure adjusted, will
be within 'OOOOOOI of its true value.
In Example 2 the differences do not at first sight appear to run
regularly, but this is only due to the fact that the final figure in
each value of 11 represents, as explained in the last paragraph, an
approximation to the true value. The general principle on which
we proceed is the same; but we use the actual difference corresponding
to the interval in which the value of x lies. Thus for x=7-41373
we should have 14=~86982-+-(~373 of 58) =-87004; this result
being correct within -oooo1.

2. Interpolation by Second ])1 /feremes.-If the consecutive first differences of 11 are not approximately equal, we must take account of the next order of differences. For example:- Example 3.-(14=log mx).

x. 14. 1st DER Y vD2nd Diff.

Q'0 77315

+713

6-1 -78533 -I2 1

1 +706 1

6-2 ~79239 - II

+695

6'3 '79934 II

+684 >

6~4 -80618 - 1 1 Q

+673

6-5 -81291

~A < .... , . .v ~ ., . .. ...1 In such a case the advancing-dzjference formula is generally used. The notation is as follows. The series of values of x and of 14 are respectively x0, xl, x2, . .and 140, 141, 142, .; and the successive differences of 14 are denoted by A11, AQ14, Thus A140 denotes 141-140, and A2140 denotes A141-A140=140-2141-l-140. The value of x for which 14 is sought is supposed to lie between x0 and xi. If we write it equal to x0+6(x1-x0) =x0-|~9h, so that 0 lies between 0 and 1, we may denote it by xg, and the corresponding value of 14 by 140. We have then

- - -)

140 = u0+0A140-%QA21¢0+ iA3140- (I). Tables of the values of the coefficients of A2140 and A3140 to three places of decimals for various values of 0 from 0 to I are given in the ordinary collections of mathematical tables; but the formula is not really convenient if we have to go beyond A“140, or if A2140 itself contains more than two significant figures. To apply the formula to Example 3 for x=6-277, we have 9=~77. so that 1¢0=-79239+<-71 of 695)"('089 Of -11)=-792s9+ 535 154-0 98='79775-Here,

as elsewhere, we use two extra figures in the intermediate calculations, for the purpose of adjusting the hnal figure in the ultimate result.

3. Taylors Theorem.-/Vhere differences beyond the second are involved, Taylor's Theorem is useful. This theorem (see IN-FINITESIMAL CALCULUS) gives the formula

H2 Q

110-1f0+C16“, ~C22+C33!+ . (2),

where, ci, cg, 60, . . are the values for x=x0 of the first, second, third, . . differential coefficients of 14 with regard to The values of ci, Q, . . can occasionally be calculated from the analytical expressions for the differential coefficients of 14; but more generally they have to be calculated from the tabulated differences. For this purpose central-difference formulae are the best. If we write 116140 = 5<A1l0 +A14,) .

52140 =A2u

- 53140 =§ 1§ '(A3'l¢ .1'f'A31»£ 2

&c.

so that, if (as in §§ I and 2) each difference is placed opposite the space between the two quantities of which it is the difference, the expressions ézllq, 64140, denote the differences of even order in a horizontal line with u0, and 146140, /463140, denote the means f

0 the differences of odd order immediately below and above this line, then (see D1FFERENc15s, CALCULUS off) the values of cl, cg, are given by

61 =I»¢51/lo- fsM53%o 'l':110M55110 “' i'iul#571ff0°l” - — Q = 62140 - 31264140-¥01, ,66110 °" sf 658110 -l- . .. c0 =/.46“140- i/165140-P i§ 0, LL57'Lf0 — . C4 = 151110 - 2566140 +15;g§ 8M0 " . . . c5=;465140-g/467140-P . (4).

50 = 56110 - }63140 -l~ . .

If a calculating machine is used, the formula (2) is most conveniently written

M9 = 1»io'l'Pi0

P1 2 C1 rf* P20

IT2:T1Z'l"fP30

Using 0 as the multiplicand in each case, the successive expressions PS, P2, Pl, 140 are easily calculated. As an example, take 14= tan x to five places of decimals, the values of x proceeding by a difference of I°. It will be found that the following is part of the tablew-Example 4.-~(14 = tan x).

x. 14. 1st Diff. 2nd Dinh 3rd Diff. 4th Diff. ~l- -l- -lr -l-65°

2-14451 732 I6

IO153 96

66° 2-24604 828 19,

10981 115 1

67° 285535 943 I3

To find 14 for x =66°23/, we have 0 =23/60 = -3833333. The following shows the full working: in actual practice it would be abbreviated. The operations commence on the right-hand side. It will be noticed that two extra figures are retained throughout. 110. /J-5140. 691/0, /463140. 64110, 2.24004 +IO§ 67U'; -§ 8¢8"') +165-5" +1900 A 175. 153

c1=4~10540“2 c2=+826'f c3=i;ro;5" f4=;;$ P19='i'4!05“7 § P29='i' 16102 }, P39=+1371 § 1:49=-l- 152 146==2.28710 Pl:-{~ro710*" P2 == %8401¢' pa: 1.10732 The value 2~2870967, obtained by retaining the extra figures, is correct within -7 of -ooooi (§ 8), so that 2-28710 is correct within ~OO0OI I.

In applying this method to mathematical tables, it is desirable, on account of the tabular error, that the differences taken into account in (4) should end with a difference of even order. If, ag. we use 1.163140 in calculating cl, and 10, we ought also to use éqlfg for calculating cz and 60, even though the term due to 64140 would be negligible if 64140 were known exactly. 4. Geometwcal and Algebmicril I7Zf€7'f7l'€f(lfZ1071'.* II1 applying the principle of proportional parts, in such a case as that of Example I, we in effect treat the graph of 14 as a straight line. We see that the extremities of a number of consecutive orrlinates lie approximately in a straight line: ie. that, if the values are correct within isp, a straight line passes through points which are within a corresponding distance of the actual extremities of the ordinates; and we assume that-this is true for intermediate ordinates. Algebraically we treat 14 as being of the form A-l-Bx, where A and B are constants determined by the values of it at the extremities of the interval through which we interpolate. In using first and second differences we treat 14 as being of the form A+Bx-l-Cxz; 1.12. we pass a parabola (with axis vertical) through the extremities of three consecutive ordinates and consid th h'. ' ' °er

at t 1s 1s the graph of 14, io the degree of accuracy given by the data. Similarly in using differences of a higher order we replace the graph by a curve whose equation is of the form 14=A-l-Bx-l-Cxz-l-Dx3+ . . The various forms that interpolation formulae t kf d ' ' ' ' ' ~

a e are ue to the various principles on which ordinates are selected for determining the values of A, B, C . . B. Inverse Irzterpololiozz.

5. To find the value of x when 14 is given ie to find the val f 1 ~ - ue o

6 when 149 is g1V€1l, we use the same formula as for direct inter I

po ation. but proceed (if differences beyond the first are involved) by successive approximation. Taylor's Theorem, for instance, gives

9 A (149-140) T(ci-I-12%-% . .)

=(149*1.f1))+P|