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To find the differences of tan Ug and of tan-uz:

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From the above, or independently, it is easily shewn that

sin a

A tan asx =

cos ax cos a (c+1)


a A tan- ax = tan ?


1 + aʻx + aʻx Additional examples will be found in the exercises at the end of this chapter. 4. When the increment of ~ is indeterminate, the opera

A tion denoted by


merges, on supposing Ax to become infinitesimal but the subject-function to remain unchanged,

d into the operation denoted by The following are illus

dxo trations of the mode in which some of the general theorems of the Calculus of Finite Differences thus merge into theorems of the Differential Calculus.

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And, repeating the operation n times,

Ax + +π
(2 sin 4 Ax)"sin ( 3c+n
A" sinc


....(1). (4x)"

It is easy to see that the limiting form of this equation is

d" sin x
sin (a +



a known theorem of the Differential Calculus.

Again, we have


axtar — a


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And hence, generally,
Alla aad


....... (3). (Ax)" AX Supposing Ax to become infinitesimal, this gives by the ordinary rule for vanishing fractions dam


da" = (log a)" at

But it is not from examples like these to be inferred that the Differential Calculus is merely a particular case of the Calculus of Finite Differences. The true nature of their connexion will be developed in a future chapter.


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Expansion by factorials. 5. Attention has been directed to the formal analogy between the differences of factorials and the differential coefficients of powers. This analogy is further developed in the following proposition.

To developed (w), a given rational and integral function of x of the mth degree, in a series of factorials. Assume $(x) = a + bx + cx12 + d.x2)

(1). The legitimacy of this assumption is evident, for the new form represents a rational and integral function of x of the mth degree, containing a number of arbitrary coefficients equal to the number of coefficients in ¢ (x). And the actual values of the former might be determined by expressing both members of the equation in ascending powers of x, equating coefficients, and solving the linear equations which result. Instead of doing this, let us take the successive differences of (1). We find by (2), Art. 2,

A0 (x) = 6 + 2cx + 3dx1) ... + mholm–1) ...... (2), 4$ (x) = 2c +3.2dx ... tm (m-1) hwm-2)... (3),

A"! (x) = m (m – 1) ... 1h

.(4). And now making x= () in the series of equations (1)...(4), and representing by Δφ (0), Δ'φ (0), &c. what Δφ ), Δ'φ(α), &c. become when a = 0, we have

Φ (0) = α, Δφ (0) = 6, Δ'φ (0) = 2c,

AMP (0)=1.2... mh.
Whence determining a, b, c, ... h, we have

Δ'φ (0)
Φ) = φ (0) + Δφ (0) +

2012)+°)+ &c. 5).

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2.3 If with greater generality we assume

$() =a + bx + cx (oc h) + dx (x h) (wc 2h) + &c.,


ing of


we shall find by proceeding as before, (except in the employ

for A, where Ax=h) $ (x) = {$ (x)} +

$4$ (a)? $A’$ (x) x (x – h)

XC +
Ax 7 (4x) 1.2
S48® (x)] x (oc h) (x 2h)

+ &c. ...(6), (A.x)" )

1.2.3 where the brackets {} denote that in the enclosed function, after reduction, wc is to be made equal to 0.

Maclaurin's theorem is the limiting form to which the above theorem approaches when the increment Ax is indefinitely diminished.

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+ &c.,

General theorems expressing relations between the successive values, successive differences, and successive differential coefficients of functions. 6. In the equation of definition

Auz = Ux+1 Uz we have the fundamental relation connecting the first difference of a function with two successive values of that function. Taylor's theorem gives us, if h be put equal to unity,

dur. 1 d'un

1 d'un Ux

+ + dec

2 da? 2. 3 daca which is the fundamental relation connecting the first difference of a function with its successive differential coefficients. From these fundamental relations spring many general theorems expressing derived relations between the differences of the higher orders, the successive values, and the differential coefficients of functions.

As concerns the history of such theorems it may be observed that they appear to have been first suggested by particular instances, and then established, either by that kind of proof which consists in shewing that if a theorem is true for any particular integer value of an index n, it is true for the next greater value, and therefore for all succeeding values ; or else by a peculiar method, hereafter to be explained, called the method of Generating Functions. But having been once established, the very forms of the theorems led to a deeper conception of their real nature, and it came to be understood that they were consequences of the formal laws of combination of those operations by which from a given function its succeeding values, its differences, and its differential coefficients are derived.

7. These progressive methods will be illustrated in the following example.

Ex. Required to express wrth in terms of U, and its successive differences. We have

Ux+1=Uz + Au;
:: Uz+2 ="U. + Au, +A(u+ Aux)

= uz + 2A4, + A'u, Hence proceeding as before we find

Ux+3 = U, + 3Au: + 3A%u; + A'uz. These special results suggest, by the agreement of their coefficients with those of the successive powers of a binomial, the general theorem

n (n − 1) Uz + nAur +

1.2 n (n − 1) (n − 2)

ARU, + &c............. 1.2.3

... (1).


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Suppose then this theorem true for a particular value of n, then for the next greater value we have

1) Ux+n+1 = Uz + nAu, +

Δ'u, 1.2

n (n


n (n − 1) (n − 2)

A%uz+ &c.

1.2.3 + Au, + na'u,+

n (n − 1)

A'u, + &c.

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