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$ (E") 0* = n* ¢ (E) 04. [ 4. Shew that, if m be less than r,

{1+ log E}"0" =» (r- 1) ...... (r- m +1). 5. Express the differential coefficient of a factorial in factorials. Ex. cm).

6. Shew that

A "O", A"On+1

form a recurring series, and find its scale of relation.

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U.2

A’u
u, + 4.2+ + &c. = 6* u, + xAu, + +
1.2

1.2

What class of series would the above theorem enable us to convert from a slow to a rapid convergence ?

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and hence calculate the first four terms of the expression.

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(P,A'-P,48 + &c.) Om = 0 if m> 2. Prove that

{log E}" Om = 0,

unless mrn when it is equal to n.

11. Prove that

1
* + nac

=

= xl-2) + (1 - n)a* + (1 - n) (2-n) x-*+&c.

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13. If Aux, y = Ux+1, 4+1 – Ux, y, and if A"ux,y be expanded in a series of differential coefficients of Ux,y: shew that the general term will be

Δ" Opta

APOP X A'O? " dye 14. Express A"x" in a series of terms proceeding by powers of x by means of the differences of the powers of 0.

By means of the same differences, find a finite expression for the infinite series

dx dy

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where m is a positive integer, and reduce the result when

m=4. 15. Prove that

F (E) a*px=a*F(aE) $x,

(2A)""Uz = (x +n - 1)(n) Auz,

f (xA) (2E)”un = (-E)" f(«A + m)Ux and find the analogous theorems in the Infinitesimal Calculus.

16. Find u, from the equations

1-11 (1) Guz

;

2t (2) Gu, = f (). 17. Find a symbolical expression for the oth difference of the product of any number of functions in terms of the differences of the separate functions, and deduce Leibnitz's theorein therefrom.

18. If Pn be the number of ways in which a' polygon of n sides can be divided into triangles by its diagonals, and to(t) = GPn, shew that

(t)-1 $(t) =

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n and a being positive quantities.
*20. Shew that
sin 2nx sin"

TAM (2n m)"
da

atm 2m+11(a +1) cos

2 if 2n>m >a all being positive.

* In Questions 19 and 20 A acts on n alone.

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xati

T

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21. Shew that if p be a positive integer

1.2.3 ...... 2p
dac
* (x + 4) ...... (** + 4p)

(Bertrand, Cal. Int. p. 185.)

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A" 1p+1 = (n+1) A" 1P+NA"-1", and apply it to construct a table of the differences of the powers of unity up to the fifth power.

CHAPTER III.

ON INTERPOLATION, AND MECHANICAL QUADRATURE.

1. The word interpolate has been adopted in analysis to denote primarily the interposing of missing terms in a series of quantities supposed subject to a determinate law of

magnitude, but secondarily and more generally to denote the calculating, under some hypothesis of law or continuity, of any term of a series from the values of any other terms supposed given.

As no series of particular values can determine a law, the problem of interpolation is an indeterminate one. To find an analytical expression of a function from a limited number of its numerical values corresponding to given values of its independent variable x is, in Analysis, what in Geometry it would be to draw a continuous curve through a number of given points. And as in the latter case the number of possible curves, so in the former the number of analytical expressions satisfying the given conditions, is infinite. Thus the form of the function-the species of the curve--must be assumed a priori. It may be that the evident character of succession in the values observed indicates what kind of assumption is best. If for instance these values are of a periodical character, circular functions ought to be employed. But where no such indications exist it is customary to assume for the general expression of the values under consideration a rational and integral function of x, and to determine the coefficients by the given conditions.

This assumption rests upon the supposition (a supposition however actually verified in the case of all tabulated functions) that the successive orders of differences rapidly diminish. In the case of a rational and integral function of x of the 7th degree it has been seen that differences of the n + 1th B. F. D.

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