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CHAPTER V
CENTRAL DIFFERENCES

If a series of values of u,, be given, we can interpolate to find any intermediate value by one of the methods in the preceding chapters. Where the values of the argument x proceed by unit intervals it has been shown that, on certain assumptions, Newton^'s advancing difference formula can be applied to give satisfactory results. If the value of ux were required for some value of x between x = o and x = I, it might be considered that we should obtain a better result if our knowledge of the shape of the curve extended on both sides of the values of x between which we wish to interpolate. That is to say, where we may choose any values of ux at unit intervals for our data, it might be of advantage if we could use a formula involving values such as u__3i u_₂, uo, u_I, u₂, u₃, ... rather than uo, u_I, u₂, u₃, u₄, .... By the advancing difference formula we expand u_x in terms of a given value of u_x and its leading differences. In central difference formulae we are not con-fined to leading differences; the formulae are based on differences obtained from values of u_x on either side of the origin.

There are various central difference formulae that are of use in actual practice, and the development of the better-known formulae is an exercise in the application of the fundamental principles of finite differences which will be advantageous to the student. It will be assumed that the values of the function correspond to equidistant values of the argument. For unequal intervals the method of divided differences should be adopted.

Gauss's formula.

The ordinary advancing difference expansion is

(x -

x0uo

uo +

tro

3•

x(x

—

I)(x-z)(x—3)

^p4

uo+....

...............
...............

i) =ua-{-xAup+ x (x — 0²11_₁
2!
+ [x (x — I) + x (x — 1) (x — 2)]
    2!    ₃!    3u
_L-1
    + [x (x—    3)_' (x—2)+x(x— ^I)(4~ 2) (x—3),4u-1+...
^Lx(x — I)    (x+I)x^— I) ^J    =up+x~u_p    ₂    0²11₁      A³u__I
3•
₊(x+ I)x(4~    1)(x—z) ('4U-2+A5u_2) +...,
since    x_r + ^xr+1 = (^x+ 1),_+I,
= u₀ + xLuo + ^x2^A2u_1 + (x + I)₃ A³_,u_₁ + (x + ¹)4 A4u_2 + .... This is known as the Gauss "forward^" formula.
A variant of this method is to write the advancing difference formula in the form
xlr)    xlr+1)    (x + ₁)(r+l)
xr+xr+1-(x+I)r+i, or r+    (Y+₁)I
we have
ux = u₃ + x'1!₀ + ) (A2u_1 + ^A3u_1) + ^x())
2 1    (^A3u_1 + ^A4u_1)
3 •

70    FINITE DIFFERENCES
whence, by writing A⁴u_₁= ^A4u_2 + A⁵ u_₂ and so on, Gauss's formula follows.

A proof similar to the above, in which the general term is evolved, and which depends upon the method of separation of symbols, will be found in J.I.A. vol. L, pp. 31, 32.
An alternative method of proof, depending on the basic formula for divided differences, follows the lines of the corresponding proof of Newton^'s formula for equal intervals. If on p. 61 we take arguments a, a + h, a — h, a + 2h, a — 2h, ... instead of a, a + h, a + 2h, a + 3h, ... and proceed to express the divided differences in terms of ordinary differences, Gauss's formula is at once obtained. This is seen more easily by writing down terms in the order u₀, u1, ^u—1, u₂, u_₂, ... and taking out the leading divided differences.

If we write the terms of the series and their differences thus:

^u—3
Au_₃
^A2u_3
Au_₂    A³u_₃
^u—1    ^A2u2    ^A4u—3
A11_₁    6,3 _{u_2}A⁵11_₃
U₀    A211_1    ^A4u_2
A1l_o    A³11_₁    A⁵11_2
ul    A2uo    A4_{11_1}A11₁    A³11₀
U₂    A²u_l
Art₂
u₃

it will be seen that successive differences in the Gauss formula above lie along the zig-zag line indicated. For this reason the formula is often referred to as the "zig-zag^" formula.

4. Stirling's formula.
If in Newton^'s formula we group the terms in a slightly different manner from that above, and use the relations

STIRLING^'S FORMULA. BESSEL^'S FORMULA

Alt_o = Du_₁ + 0²u_₁

^A2u0 = Q2u—1 + ^A3u—1;
.............................

0³u_1 = 0³u_2 + A⁴u_₂ ..............................
we have
ux = u₀+ x (Du_1 + A2u_1) + x2 \' ²21_1 + 0³u_—1)
+ x3 (^A3u_1 + ^L\4u—1) + ...,
which, on substituting the single function (x + ¹)r+1 for x,. + ^xr+1 and so on, becomes easily
ux = u₀+ x0u_1 + (x + ¹)2 ^O2u_i + (x + ^I)3 '^,32_2
+ (x + 2)₄ 0⁴2_2 + ... .
This is another form of Gauss^'s formula—the "backward^" form.

It should be noted that this is also a zig-zag formula. Here we take an upward step in the diagram from u₀and then proceed alternately, whereas in the forward formula the first step is downward.

Taking the mean of the two Gauss formulae we arrive at the
following expansion :
5. Bessel^'s formula.
Transforming the origin in the Gauss backward formula from o to — 1, we have
ux = u₁ + (x — I) Auo + ^x(x I) _Q2ux (x — I) (x — 2) 0³2 -1
2.    °    3¹+ (x+    ^I)^x (₄~    1) (x—2)04u1+.... The mean of this and the forward formula is
u_x= (uo + u₁) + (x — Duo + ^x⁽₂ ^I)2 (^A2u_1 + 0²u₀)+ ..., which is Bessel^'s formula.
z
ux = 2i₀ + x (Au₍₎ + O2i _1) + ₂~ A² u_1 + x (^x2 -      12) 2 (^3Z1_1 + ^A3u_2) 1 ^+x2(x² — 12) 0⁴2—2 + x (x² — 12) (x2 — 22) 2 (-SU—2 + 0⁵2_3)
4¹5¹+ x²(x² — 162 _i(x² — 22) O^su_₃ + ... .
This is known as Stirling^'s formula.

72    FINITE DIFFERENCES
6. Everett's formula.
The Gauss forward formula with interval x and initial term v₁ may be written
^vx+l = v₁ + xhv₁ + x₂0²v₀ + (^x+ I)3 0³v₀ + (x + ¹)4 ^A4v-1
+ (x + 2)₅ A⁵v_₁ + .
The backward formula with interval (x — I) and initial term v₁ gives
vx = v1 + (x — I) t v₀ + x₂0²v₀ + ^x,^A3v_1 + (x + I),1 ^L4v-1
+ (x + I)5 05v_2 +;...
Subtract the second series from the first : then, since

^vx+1 — ^vx = ^Lvx,

we have
Avx= xhv₁+(x + 1)₃ A³v₀+(x + 2)₅0⁵ v__I +...
— (x — I) Av₀^—x₃0³v_1 — (x + I)5 / 5v_2 — ....
Put ^ux, ^Dux, ^A2ux, ... for ^Avx, ^A2vx, A3vx, .... Then
ux=xu₁+ (x + I)3A2u0+ (x+z)5A4u_1+...
— (x — I) u₀ — x₃0² u_I — (x + I)₅ A⁴u_₂ — ....

When x is less than unity a convenient form of this formula for interpolation between u₀ and u₁ is obtained by putting e = — x; thus

ux — xu1 + x      (^x2- 1₎A2uo + x (x² ₅)_I (x² — 4)
A4u_1 +
3    ...
    + S^uo + (e2      ¹) A2u_₁ ,+     (e2      — I)(2    4) ^A4u—2 + ...,
    3•    5^I•
the most common form of Everett^'s formula.

The above elegant proof is due to G. J. Lidstone (J.I.A. vol. ix, p^p.349—52). In his note on this formula Mr Lidstone shows how to obtain another formula similar to the above for interpolation between u__4.and u_i . The mean of the two Gauss formulae is taken in the same

way, but x ; v₀ and x ; v₁ are used in place of x ; v₁ and x— ; ^v1 respectively : the formula then becomes

SHEPPARD^'S RULES    73
^uD ^=uD+pe2!      Duo-1(p2    4)     p2 4    4      ⁾O^su_—i+...
•
q²     _Z    4 Du_, — (q²    44     ₁^q2    L 3 u_₂ — ... ,
where p = +xandq=—x.
This form is generally known as Everett^'s ^"second^" formula; it is specially adapted for use in statistical work.
7. Sheppard's rules.

Dr W. F. Sheppard has laid down certain general principles for obtaining central difference formulae which are very simple in their application. By adopting a slightly different notation from the usual a difference table is constructed from which the formulae can be written down with little trouble. Thus :

Du_₁ is denoted by (— I, o), i ²u_₁ is denoted by (— I, o, I),

(o, ^1),^O2uo
(^I, ²); ^O2u1

(o, I, 2),
(^I, 2, 3);

Luo Du₁
^03u_i is denoted by (— I, o, I, 2),
D³uo o, I, 2, 3),
03ui (^I, 2, 3, 4) ;
and so on.
The difference table then becomes

x	u_x	Du_x	A²u_x
x_₂	^u—2
		(^—2,^— I)
x_₁	u_₁	(— I, o)	(— 2, — 1, O)
xo	u₀	(0, I)	(— I, 0, 1)
x₁	u₁		(O, I, 2)
		(I, 2)
x₂	U2

The Newton advancing difference formula may be written

^ux = ^uo + (x — ^x0) (^o, ^I) '+ ' (^xI ^x0) (^x     2      xi) (O, I, ₂) +(x—xo)(x—xi)(x—x2)(o,I,2,3)+...,
I    2    3
which is ux = uo + xzu_o + x., t ²u₀+ x₃A³u₀+ x₄⁴u₀+ ... .

74    FINITE DIFFERENCES
Gauss's forward formula:
ux = Ito + (x — x0) _(o,1)    (x I ^xo) (^x     ₂    x1) (— , o, 1)
+ (^xI ^xo) (^x     2      ^xi) (^x     3x      t) (— I, O, I, 2) -^Jr ... ,

orux=uo+x(o,i)+x2(—I,0,I)+(x+1)3(—I,0,I,2)+...

= uo + xAuo + x₂A²u__I + (x + i)₃0³u_₁ + ....
The rules are

(i)Start with any tabulated value of u_x.

Pass to successive differences by steps either downwards or upwards.The new suffix introduced at each step determines the new factor, involving x, for use in the next term.

By means of these rules the formulae of Newton, Stirling and Bessel

can be written down at once, whether for equal or unequal intervals.

8. It will be of interest to compare the results brought out by applying first, a central difference formula, and, secondly, the ordinary advancing difference formula, to the same set of data.

Example

Interpolate by means of Gauss's forward formula to find the present value of an annuity of 1 p.a. for 27 years at 5 per cent. compound interest, given the following table :

No. of years x:

20253

⁰

Annuity-value

10.3797

12'4622

⁰

939

5'37

5 16'3742 17

1591

If we take

years as the origin and

years as the unit, the value

required will be

₄

x	u_x	Du_x	A²u_x	~^3ux	Mu_x	2,⁵U_x
— 2	^10.3797
		2^.0825
— I	12^.4622		— '45⁰⁸
		1^.6317		'⁰977
0	¹4'⁰939		— '353¹		0215
		1^.2786		'0762		•⁰⁰54
I	15'3725		— _'276₉		— 0161
		1^.0017		•o6ol
2	^16.374²		— •2168
		'7⁸49
3	¹7'¹59¹

ADVANTAGES OF CENTRAL DIFFERENCE FORMULAE 75
The Gauss formula is
ux = u0 + x0u₀ + x₂4²u_₁ + (x + 1)₃ A³u_₁ + (x + _/r)4 A⁴u_2
+ (x + 2)5 A⁵11_₂ When x = .4 the successive coefficients are

•4; — •12; — •056; •0224; •010752

and to four decimal places the value of u.₄ is 14^.6430, which agrees with the tabulated value.
To apply the advancing difference formula we take 15 years as the origin and are required to find u_x when x = 2^.4.
In the formula

ux = u₀ + xiu₀ + x₂A²u₀ + x₃0³uo + x₄A⁴u₀+ x₅z ⁵u₀

the coefficients are

2^.4; 1^.68; •2²4; — '⁰33⁶; — •010752.

On evaluating the expansion we obtain for u₂.₄the value 14^.6430 as above.

It will be seen that the two results are in agreement (as indeed they must be, since the same values of zi are used), and it may be asked therefore wherein lies the advantage of using the central difference formula. This question will be discussed in the next paragraph.

9. Consider an approximation to a particular value of u_x based on, say, r values out of n available. The error in the approximation, as measured by the first neglected term, is least when the co-efficient of that term is least. It can be shown that this happens when the values of u_x upon which the interpolation is based range round the space in which x falls, so that x is as nearly as possible central. The central difference formulae give a systematic method for building up the table subject to these conditions.
Again, the central difference coefficients are as a general rule smaller than those required for the calculations in the advancing difference formula (as will be seen in the Example) and, by a suit-able choice of origin, the arithmetical work may be reduced to a minimum.

It should be noted that, in the phrase " as measured by the first neglected term," this measure is not theoretically complete; it is how-

76 FINITE DIFFERENCES

ever generally sufficient in practice if the first neglected order of differences is constant or is changing but slowly. When this is not so it will not necessarily be true that a central difference formula beginning with uo is more accurate than the advancing difference formula beginning with the same term. [See p. ₃₃₇of Sheppard's Paper, cited below.]

10. Relative accuracy of the formulae.
The relative accuracy of the various central difference formulae can be investigated in an elementary manner on the following lines. The Gauss forward formula is

u_s = up + xzu, + x₂0²1[_₁ + (x + I)3 ^Asu_1 + (^x+ 1)4 A⁴u_₂ + ... .

It is easy to show that if we expand u_s by Stirling's formula as far as a certain order of even differences we can obtain by a simple transformation the above formula to even differences. Similarly it can be proved that Bessel's formula and Gauss^'s formula are identical to odd differences. Now the Gauss formula involves only ordinary differences while the other two series involve differences of the form ₂(Aⁿu,. + ^Anur^.+1) : these may be called " mean" differences. If instead of proceeding to constant differences the series stop short at, say, rth differences—which are not constant—the use of any of the formulae will involve an error. It remains to examine which of these formulae gives the best result in different circum

The following demonstration is based on that given in greater detail by Mr D. C. Fraser in J.I.A. vol. L, p. 25.

Suppose that x is not greater than •5. Then, by calculating the coefficients in Gauss^'s formula, it will be found that for positive values of x none of the coefficients (except that multiplying ,duo) differs greatly from + .5 times the preceding coefficient. (See Table, J.I.A. vol. L, p. 25.) Thus the terms after that involving the third difference are approximately equal to

(x + ^I)4 (A4+ 20⁵+...)11_2, i.e. to (x + ¹)4 . 2 0'^4u-2 +
If therefore we substitute ₂(,⁴u_₂ + A⁴u_₁), the mean difference in line with u , for,⁴u_₂ in Gauss's formula, we make the formula

approximately correct to fifth differences, without having to

RELATIVE ACCURACY OF THE FORMULAE 77

calculate the actual coefficient of the fifth difference. The subtherefore improves the accuracy of the formula.
When, however, the substitution is made, it will be found to reproduce Bessel^'s formula to the fourth mean difference. There-fore Bessel's formula to fourth mean differences is usually more accurate than Stirling^'s to the fourth difference.
It may be shown similarly that Stirling's formula to odd mean differences is more accurate than Bessel's to the same order of differences.
The above demonstration is only approximate : a strict investigation into the relative accuracy of central and advancing difference formulae requires rather more elaborate mathematical discussion. (See Whittaker and Robinson, Calculus of Observations, p. 49; Lidstone, T.F.A. vol. Ix, pp. 246-257; Fraser, J.LA. vol. L, pp. 25-27; Sheppard, Proceedings of the London Mathematical Society, vol. Iv, Parts 4, 5^.)
Mr D. C. Fraser has given the following criteria summarising the properties of interpolation formulae:

(i)Formulae which proceed to constant differences are exact and are true for all values of n whether integral or fractional.

It should be noted that these rules, and those given in para. 7, are quite general ; they apply to all formulae based on finite differences, not ending with mean differences.11. Apart from the general superiority of central difference formulae certain of the formulae possess distinct advantages in special circumstances. For example, for the bisection of an interval Bessel's form is convenient, since the alternate terms are zero. We have, at once,

ni= ((^u0⁺III) 8{2(^A2u-1^+A2u0)] +T2F[2(A4u-2+04u-1)] —....Again, in using Everett's formula for the subdivision of intervals the terms are such that they may be used twice : they

78    FINITE DIFFERENCES
occur both in the "x^" expansion and in the "6" expansion. An example will make this clear.
Example 2.
Given    x    3⁰    35    4⁰    45    5⁰    55    6o
^ux    771    862    Tool    1224    1572    2123    2983
obtain values for u_x for all integral values of x between x 40 and x=50.

The difference table is

x	u_x	Du_x	A²u_x	^L.3ux	L'u_y
3⁰	77¹
		9¹
35	862		48
		¹39		3⁶
40	1001		84		5
		223		41
45	1224		125		37
		34⁸		7⁸
50	1572		203		z8
		55¹		106
55	2123		3⁰9
		86o
6o	2983

Everett^'s formula gives
ux=xul+(x+ 1)3 A2u0+(x+2)5/4u_1+...
+ e^uo + (S + ¹)3 ^i12u_1 + (¹5 + 2)₅ 0⁴u_₂ + ..., also ^ul+4 = 6u2 + (e + 1)3 A2 ^u1 + (e + 2)₅ A⁴% + ...
+ xu_l + (x + I), A²u₀ + (x + 2)5 0411_₁ + ...,
and the second line in u_l+is the same as the first line in u_x.

Since the data are given at quinquennial points and we require values at individual points, we may write x = •2, •4, •6, ... and 6 = •8, •6, •4, ... . The first line of u.₂ will be the same as the second line for u₁.₈and so on.

The coefficients of the terms in the first line of the formula for u_x are, to fourth difference
,

	•2 — •032 ^.006336
	.4 — •056 •010752
	•6 — •o64 •x11648
	•8 — •048 ^.008064

*x xu*		x (x² 1) O^zuo	x (x^a - 1) (x^$ - 4) Al _{u_1}	Sum of firstthree terms	Sum of second three	Inter polated result
	¹	3	51	(ii) + (iii)	terms	(v) + (vi)
(i)	(ii)	(iii)	(iv)	+ (iv) (v)	(vi)	(vii) _-I

2	200^.2	2'7	0^.0	197'5
4	4⁰⁰'4	-4'7	0^.1	395'⁸
•6	600^.6	-5'4	0^.1	595'3
•8	800^.8	-4^.0	0^.0	796^.8
•2	244^.8	-4^.0	0'2	241^.0	796'8	1037^.8
•4	489'6	-₇.₀	⁰'4	4⁸3'⁰	595'3	¹⁰7⁸'3
⁶	734'4	-8•o	0^.4	7²⁶'⁸	395'⁸	1122^.6
•8	979^.2	-6•o	0^.3	973'5	¹97'5	I171^.0
2	314'4	- 6'5	0'2	308'1	973'5	1281'6
•4	6z8^.8	-11^.4	0^.3	⁶¹7'7	726^.8	¹344'5
.6	943'²	-¹3'⁰	⁰'3	93^0.5	483^.0	1413^.5
•8 j 1257^.6		- 9^.7	0^.2	1248^.1	241^.0	1489^.1

The only column which needs explanation is column (vi). This column represents the second set of three terms of the formula, correct to fourth central differences, and is obtained by writing down in the reverse order the values of the previous column applicable to the sum of the first three terms.
It should be mentioned that the values in column (vii) of the table do not quite agree with the tabular values: the tabular values are 1038, 1081, I122, I172, 1281, ¹345, ¹4¹5, 1490. The reason for this is that the function upon which the original values depend is not a rational integral function of the independent variable and that therefore a formula based on finite differences is only an approximate representaof the function. The example is based on the rates of mortality according to the H" table, the data being 10⁵ times the probability of dying in the year of age x.
12. Just as 0 = E - 1, or Du_x = u_x+₁- u_x, similar symbolic identities may be deduced from the relations existing between u_x, ux_+i and ux_i. Dr Sheppard has introduced the following
notation, which is widely used by mathematicians:
(Sux = ux+7l - ^ux_i,

and µu_x = (^ux+4 + ux_i).
USE OF EVERETT^'S FORMULA

The work is best arranged in tabular form, thus:

8o    FINITE DIFFERENCES
The relationships between E, 8 and are quite easy to establish. 8 - Ei — E-i_E^-i[E — 1] -E^-iA. :. 82n E-n _A2n_.
Also    µ . 2 (El + E-i),
and    µ8 = .4 (E — ^E-1) = (AE-¹ + A),
µ82n+1    [E-(n-1) — E-(n+1)] A2n
(^AE—1+ A) ^{A2n I.—n}•
Again    2µ 2E1 — 8,
or    Ei    + 28.
By means of these symbols the central difference formulae can be written down in very convenient form. For example, Gauss's forward formula is    44
ux = 11_°+ x8u_i + x₂8²u₀ + (x + 1)3 8³u_i + (x + 1)4 O⁴u_°+ ..., and Stirling^'s becomes
^ux = 11₀+ xi8u₀ + x²8²u₀ + (x + I)3 i8³u₀ + ....

EXAMPLES 5

r. Apply a central difference formula to obtain u , given u20 = 14, ¹¹24⁼3²,^u28⁼35,¹¹32⁼4^0.

Given u₂ = 10,11₁= 8, u₀ = 5,11_₁= Io, find u_i by Gauss's forward formula.

Use Stirling

s formula to find

_28i

given u

₂₀

= 49

²²

¹¹

= 48316,

¹¹

= 47

⁶

¹¹

= 45926,

u40

= 443

^06.

Given

2854,

u24

= 3162,

1120

= 3544,

U32

= 3992;

^find

_u25

by Bessel

s formula.

₂₁

18'4708; ay,r = 178144;

₂₉

1070; a

₃₃

¹⁶

'343

;

37 =

5'5

Find a„ by Gauss's forward formula.

What are the practical advantages arising from the use of central differences in interpolation?

Employ Stirling

s formula to obtain successive approximations to (1'02125)

⁵⁰

, given

(

_oi

)5°

= r64463;

(

₁

₀₂

)50

^2.6

59; (

^1.0

⁵⁰

= 4'3

⁸

;

015)

⁵⁰

2'10524; (

⁰²

⁵⁰

= 3'437

^11.

EXAMPLES 81
7. Find formulae true to third differences for the bisection of an interval

(i)in terms of the two nearest values of the function and of differences of the functions ;
(ii)in terms of values of the function only.

Apply either formula to find P

_35i

given the values of P

at 20, 30, 40, 50 to be

, 3493 respectively.

8. Given the table

31032033

⁰

360 log x 2

4914 2'5052

^2.

¹⁸

'53

'544

'55

⁶

3 find the value of log

₃₃₇₅

by a central difference formula.

9. Prove that if third differences are assumed to be constant

>_2

u_x=x11₁+^x ^x_b-- ^I~A²uo+yuo+^yy ₆¹A²u__I,

where

y =

1 —

Apply this formula to find the values of

_u11

to u

₁₄

and u

₁₆

to u

₁₉

given

that

₀

= 3010,

₅

2710,

₁₀

2285,

₁₅

1860,

₂₀

1560,

1125

1510

and u

₃₀

¹⁸

so. From the table of annual net premiums given below find the annual net premium at age 25 by means of Bessel's formula:

Age Annual net premiums

20•0142724•0158128•01772

⁰¹

⁶

I1.

Apply a central difference formula to find

f (32),

given

(

= '

⁰

)

= •

3027,

(35) = '33

⁸⁶

⁰

)

'3794•

12.Use Gauss^'s interpolation formula to obtain the value of f (41),

given

⁰

) = 3

⁶

⁸

(35) =

995

^.1

⁰

)

2400

(45) = 1876

(50)

1416

Verify your result by using Lagrange

s formula over the same figures

13.Buchanan^'s formula for interpolating between uo and u₁ is 11.₂ = •8u₀ — •o48A²u_₁ + •0128A⁴u_₂ + •211₁ — •0320²11₀ + •oox6A⁴u_₁ u.₄ = •6uo — •o64A²u_₁ + •01444⁴11_₂ + •411₁ •o56A²uo + •oo8o0⁴u_₁ U.₆ = •411₀ — •o56A²u_₁ + •oo8o4⁴u_₂ + •6u_i — •0644²11₀ + •O1444⁴u__I 11.s = •211₀ — •032A²u_₁ + •oo16A⁴u_₂ + •8u₁ — .048 \²uo + '01284⁴11_₁. Prove the formula for u.₂.

82 FINITE DIFFERENCES

Show that any central difference formula can be developed from Lagrange. Apply a central difference formula obtained thus to find f (32), given that f (2) = 2'626; f (3) = 3'454; .f (4) = 4'784 and .f (5) = 6^.986.

Given

uo, u

, u

₂

₃

₄

₅

(fifth differences constant), prove that

25(c—b)+3(a—c)

2i = zc +

₂₅₆

where

a=uo+u

₅

b=u

₄

;

⁼

u2+u3.

i6. A series is formed by the division of the terms of the two series

624 ...

n !

420120840 ...

(n + 3)!

Obtain an interpolated value for u

_2I

_2}

of the new series by a central difference formula and compare the result with the quotient of

u2i.

by v

_2I

in the component series.

17. The following is a difference table written down in Woolhouse's notation :

a_2

-I

₂

Tf a

(a_

)

and c

= (c_

₁

), show that Stirling's formula (to fourth differences) can be expressed as

+ Dx

⁴

where

B, C, D

are functions of a

bo, c

do only.

Prove that in Woolhouse's notation

₁

₂

₃

c, +

₄

do correct to fourth differences.

Show that the sum of the terms of the series u_₂, u_₁, uo, u_I, u₂can be expressed in the following form

Auo