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and apply the theorem to demonstrate that

) Unt0

JC (oc — 1) (x + 1) XC (30— 1) (xc— 2)
A®Um_, +

Aup, + &c. 3

2 (C+1)
(2) Un = Un + Aunt

2 (x - 1) (C+1) 2 (22 — 1) (x+2)

A'un + &c. 1.2.3



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8. Shew that the function

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+ &c.

t-a (t-a) (t-5)
1+ +

b (a - b)(a -c)
becomes unity when t=a, and zero when t=b, C,

deduce Ex. 6 therefrom.
9. Demonstrate Stirling's Interpolation-formula


t (t° – 1)
Uz = x + 5 A (u'+u_) +
, (

Aux +

A8 (u_, +1_) 1.2

ť (to — 1) +

A'u_, + &c.

(Smith's Prize, 1860.) 10. Deduce Newton's formula for Interpolation from Lagrange's when the values are equidistant.

11. If u radii vectores (u being an odd integer) be drawn from the pole dividing the four right angles into equal parts, shew that an approximate value of a radius vector (u.) which makes an angle 8 with the initial line is

sin (0-a)



sin 3 (0 - a)

where a, b, ... are the angles that the li radii vectores make
with the initial line.

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12. Assuming the formula for resolving

f (x)

(x a) (x b)... (3C k) into Partial Fractions, deduce Lagrange's Interpolationformula.

13. If $ (x) = 0 be a rational algebraical equation in a of any order, and 2, 7, ...Zn be taken to represent • (1), $ (2),... (k), find under what conditions

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r=1 %v (2, — )... (22 – 2x) may be taken as an approximate root of the equation.

14. Demonstrate Simpson's rule for finding an approximate value for the area of a curve, when an odd number of equidistant ordinates are known, viz.: To four times the sum of the even ordinates add twice the sum of the odd ones; subtract the sum of the extreme ordinates and multiply the result by one-third the common distance.

15* Shew that Simpson's rule is tantamount to considering the curve between two consecutive odd ordinates as parabolic. Also, if we assume that the curve between each ordinate is parabolic, and that it also passes through the extremity of the next ordinate (the axes of the parabolæ being in all cases parallel to the axis of y), the area will be given by


15 (. +yn) – 4 (y, + Yn_s) + y2 + yn_2

Area=h xy - 24

167. Given ur and

Untu , and their even differences, shew that 1 1.3 1.3.5

A® + &c. uz++1 8.16 8.16.24

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AP +


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On the comparative merits of these and similar methods see Dupain (Nouvelles Annales, XVII. 288).

+ The notation in this formula (due to Gauss) is that referred to on the top of page 56,

17. Shew that

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< (v + 2r – 1) d'une «(x+3r. – 1) (x + 3r — 2) AUn-84+&c.

n(n+1) Antally meet&c



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In what cases would the above formulæ be especially useful ?

18. Shew that the coefficient of A’un in (27) is equal to


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and hence shew the exact relationship in which (27) and (18) stand to each other.

19*. If from the values Wa, Up... of a function corresponding to values a, b, c ... of the variable, we obtain an Interpolation-formula, Aug=4, +B(c - a) + (c - a)(z 6)+P(c - a) (x - 5)(cc)

+ &c., shew that Au, AB



d where A4 (a, b, ...) = 0 (b, c, ...)-° (a, b, ...).

Deduce (2), page 35, from the above formula.



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* Newton's Principia, Lemma v. Lib. III. This is the first attempt at finding a general Interpolation-formula, and gives a complete solution of the problem. The result is of course identically that obtained by Lagrange's formula, though in a very different form.




1. THE term integration is here used to denote the process by which, from a given proposed function of x, we determine some other function of which the given function expresses the difference. Thus to integrate u, is to find a function V, such that

Δυ, = μ. The operation of integration is therefore by definition the inverse of the operation denoted by the symbol A. As such, it may with perfect propriety be denoted by the inverse form A-7. It is usual however to employ for this purpose a distinct symbol, £, the origin of which, as well as of the term integration by which its office is denoted, it will be proper to explain.

One of the most important applications of the Calculus of Finite Differences is to the finite summation of series.

Now let Uy, u,, wg, &c. represent successive terms of a series whose general term is Ux, and let 2*2 = Ua+Watu+Wa+2...+U9-1

(1). Then, a being constant so that Un remains the initial term,

we have


Un + Watı + + Uxmat Uxo..... .(2) Hence, subtracting (1) from (2),

Δυ, = u» ... υ = Δ'u. It appears from the last equation that A-2 applied to Ug expresses the sum of that portion of a series whose general term is Ux, which begins with a fixed term u, and ends with Urs On this account 4-7 has been usually replaced by the symbol £, considered as indicating a summation or integration. At the same time the properties of the symbol E, and the mode of performing the operation which it denotes, or, to speak with greater strictness, of answering that question of which it is virtually an expression, are best deduced, and are usually deduced, from its definition as the inverse of the symbol A.

Now if we consider Euz as defined by the equation

Euz =Uz-; + Uz-+ + Uq....... (3), it denotes a direct and always possible operation, but if we consider it as defined by the equation Συγ = Δ'u,

.(4), and as having for its object the discovery of some finite expression vx, which satisfies the equation Av. = Ux, it is interrogative rather than directive (Diff. Equat. p. 376, 1st ed.), it sets before us an object of enquiry but does not prescribe any mode of arriving at that object; nor does it give us the assurance that there is but one answer to the question it virtually propounds. A moment's consideration, indeed, will assure us that the number of expressions that can claim to be denoted by A2x is infinite, since it includes the quantity

+ Ux_1?

Uat Watt whatever value a may be supposed to have, provided only that it is one of the series of integral values which « is supposed to take. We cannot therefore consider the definitions of Eu, contained in (3) and (4) as identical, and shall therefore proceed to investigate the relation between them and the restrictions as to the use of each.

It is obvious that the Eu, of (3) is one of the functions represented by the Aux in (), since it satisfies the equation Av=Uz But this is of no value to us unless we can recognize to which of the functions represented by Aou, in (4) it is equal, or obtain an expression for it in terms of any one of them. This last we shall now proceed to do.

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