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Essays on Powers and their Differences. By Francis Burke, Esq. of Ower, in the County of Galway, Barrister at Law, and A.B. of Trinity College, Dublin.-After the numerous attempts that we have seen, for the demonstration of this celebrated theorem, and the various objections that have been urged against some of the principles on which they have been made to depend, it seemed almost hopeless to expect to meet with one which should entirely avoid these or other difficulties. We confess that under a feeling of this kind we began the perusal of the present essay; and certainly the preliminary part of it by no means impressed us with a more favourable idea: yet, on farther examination, we have no hesitation in stating it as our opinion that it depends on principles wholly unobjectionable, and that we can see no reason for desiring any thing more satisfactory. This remark, however, must be understood as applying to that part only in which the law of the co-efficients is derived; for we do not admit the legality of the preceding part, in which the author proves the law of the indices, viz. that the expansion

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of (1+x) is of the form + C + D x3 +, &c.

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Yet, as this part has been demonstrated on undeniable principles, we do not consider a deficiency here as materially affecting the general conclusion. From this point, at which the defect of most demonstrations begins, we may follow the author with great pleasure and satisfaction; and, though we cannot give the demonstration at full length, we will endea vour to present such an abstract of it as will enable the reader to comprehend the principles on which it is made to depend,

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Admitting the form above stated, viz. that, (1+x) =1 + = x + Cx2 + Dx' +, &c., it follows, by means of the

usual transformation, that (p+x)

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= P

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+ Cx p +, &c. and the object of the author is to determine the law of the several co-efficients C, D, &c. which he does by means of the following proposition and its corollaries.

- Prop. If o, p, q, r, s, &c. are the terms of an arithmetical series, whose common difference is d, then will p+d,q + d, r+d, s+d, &c. be respectively equal to the corresponding terms of the series, o, p, q, r, s, &c. If we take the successive differences of theth powers of those terms, we shall have for the first

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term

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term of the nth order of differences, the series d(m

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d (———2) d ( ————3) d, &c. ( —n—1) d× wn.

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This is demonstrated as follows:

Powers.

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t =, &c.

Whence taking the first, second, &c. differences, and observing that, op+d, p = q +d, q=r+d, &c. we have (retaining only the first term of each) the following results: Second Differences.

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First Differences.

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In the same way, taking the differences of the second differences, we have,

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#d() d(-)d (7–3) d's
1)

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+, &c.

Hence, by an obvious and infallible deduction, the first term of the nth differences will be

d()() d, &c. (7———7—1) d w

--

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Q. E.D.

from which proposition, the author deduces the following corollaries, viz.

1. If, instead of the preceding series, we take the series *+o, x+p, x+q, &c., and arrange the several differences according to the several powers of x, the first term of the nth differences will then become

d ( — — 1 ) d ( —
— — — 2 ) d ( 7

C)(3) d, &c. (1) da

and which will be the constant first term of the several nth differences.

2. If=n, an integer, then the above becomes

n (n−1) (n—2), &c. 3. 2. 1. d", or = 1, 2, 3, &c. n.d"; which latter expression not only represents the first term of the nth differences of nth powers, but exhibits the entire differences, which are therefore constant, and consequently the (2+1)th differences are all equal to Zero.-We do not attempt to give here the demonstration, or rather the explanation, of these deductions; because it would occupy too much room, and because we conceive that it is sufficiently obvious. Perhaps the author himself has been too prolix in this respect.

The demonstration of the binomial theorem is drawn immediately from the two preceding corollaries, by substituting the series of natural numbers 1, 2, 3, &c. instead of q, in the general expansion,

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wards the least.

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+, &c. viz. proceeding in an inverted order, to

(x +4)

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+, &c.

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(x+2)=x+b.2.x +

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r = x + b.x. x

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+, &e. t being taken to represent the co-efficient of the #+ Ith term in

the expansion of (x+q) and the form of which is required. Now, in the above series of arithmeticals, the common difference d= 1; therefore, the first term of the nth differences is equal to

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which is obviously the same as the nth differences of the latter terms, or of the n+1th terms of the expansion, and which from

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cor. 2 is equal to 1. 2. 3. 4....n.tx r ; whence, equating these two, we have for the value and form of t (that is, of the co-efficient of the 7+1th term of the expansion,) t =

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We are aware that this demonstration may be too much contracted to appear quite perspicuous to some of our readers: but it must be remembered that we profess only to explain the principles of it; and not to give the demonstration itself, which appears to us to be perfectly satisfactory.

The author concludes this essay by a demonstration of the Multinomial Theorem of De Moivre: but we cannot enter on any explanation of it in this place.

Essay II. Differences.

On finding per Saltum the several Orders of

The formulæ for finding per saltum the several orders of differences in series, the author observes, have been usually deduced from a repeated arithmetical subduction. Thus, if the successive quantities are a, b, c, d, &c. then

Ist diffs. a-b, b-c, c-d, &c.

2d diffs. a-2b+c, b—2c+d, &c.

- 3d diffs. a-3b+3c―d, &c.

and universally, the nth differences, having their several co-efficients the same as those of a binomial raised to the nth

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powers, will be represented by the formula a-nb + __n (n−1) (n−2) d +, &c.

I. 2. 3.

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The object of the present essay is to generalize these formule; and, as Mr. Burke states, to enlarge the analogy which has been observed in some cases to prevail between differences and fluxions. We cannot attempt to enter into the particular process of the author: but, as some of his propositions contain new properties, and others considerably extend old properties, it may not be amiss to point them out to the reader's notice; though for their demonstrations we must refer to the Essay itself.

1. The th differences of the nth powers of quantities in arithmetical progression are constant, and equal to 1. 2. 3. 4. ... n. d", d ́ being the common difference. This important property, we have seen, the author deduces as a corollary in the preceding essay, and it is only repeated here to preserve an uniformity in the present chapter. The invention of this for mula is due to Briggs, he having been led to it in the construction of his logarithms: but its demonstration has been generally made to depend on the fluxional or differential calculus; whereas Mr. Burke, by making its demonstration rest on first principles, inverts the order of proceeding, and therefore renders the latter dependent on the former.

2. If there are n number of arithmetical series, whose common differences are respectively a, b, c, d, &c.; let all the different corresponding terms of the several series be multiplied together, the nth differences of the products will be constant, and equal to 1. 2. nxabcd &c.'—

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n + 1 (n + 1) (n + 2) (n + 1 ) (n + 2) (x+3) I. 2. 3.

1.

2.

3. The series 1, &c. will have the nth differences of its terms common, and equal to unit, which is the definition of triangular (figurate) numbers of the nth order.'

4. If the qth powers of the natural numbers, and the rth powers of the corresponding triangular numbers, which admit two orders of differences, and the corresponding sth powers of the triangular numbers having three orders, &c. be all multiplied as follows:

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these products will have as many orders of differences as there are units in g + 2 r + 3s+, &c. and the last of these will be con stant, and

REV. MAY, 1814.

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