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denote any As m may

Cor. I.

number integral or fractional, positive or negative; hence the division, involution and evolution of a binomial, may be performed by this theorem.


Ex. I.

Let m=, then a+b+=a+ ta t+X-Xa-ke+, &c. being applied to the extraction of the square root of a2 + x2 (by inserting a? for a and *2 for b) the same series results as formerly, (Chap. IV.).

Ex. 2.


If I is to be turned into an infinite series, fince <=1X1–1 %, let a=1,b=-r, and m=-1; and the same feries will arise as was obtained by division. (Chap. I.).


In like manner, vzrk=z=r? x 2rz— may be expressed by an infinite series, fupposing a=2rz, b=-z?, and m=- , and then multiplying that series by 32.


Cór. ż. This theorem is useful also in discovering the law of an infinite series, produced by division or evolution. Thus, the series expressing the square root of a2 + x?, consists of a, together with a series of fractions, in the numerators of which are the even powers of x, and in the denominators the odd

powers of a ral coefficients of the terms of the whole series, as deduced by the theorem, will be:

The nume

1 I, +

2 XI

IXI 2.2 X 1.2


IX1.3 2.2.2 X 1.2.3


&c. the point being used (as x' it often is) to express the product of the numbers between which it is placed. The law of continuation is obvious, and the feries

may be carried on to any number of terms, without using the theorem. Hence also the coefficient of the nth term is I*1.3.5 &c. .. (R-2 terms)

; and it is t, if n * 1. 2. 3. 4. &c. (121) is an even number, and --, if n is odd.

Note. If the binomial is a +b, the signs of the terms of any power are all positive: If it is ab, the alternate terins are negaS



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tive, beginning at the second. This theorem may be applied to quantities which consist of more than two parts, by fupposing them distinguished into two, and then substituting for the powers of these com-pound parts, their values, to be obtained also, if required, from the theorem. Thus, a+b+c=a +6+


An infinite series may itself be multiplied or divided by another; it may be involved or evolved, and various other operations may be performed upon it which are necessary in the higher parts of algebra. The methods for finding the sum depend upon other principles.

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IV. Properties of Numbers.

Theor. I. The sum of two quantities multiplied by their difference is equal to the difference of their squares.


Let the quantities be represented by a and b, then a +bxa-b=a-6, as appears by performing the operation.

COR. If a and b be any two quantities of which the sum may be denoted by sg the difference by d, and their product by po then the following propositions will be true.

1. a? +b=s-2p.

2. a?-basd. 3. a3 + b3 = 33–3ps. 4. a3-63=59ddp. 5. a4+b4=54-4ps+2p% 6. 24-64553d--2sdp, &c.

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It is unnecessary to express these propositions in words, and the demonstrations are very easy, by raising a+b to certain powers, and making proper substitutions.

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THEOR. II. The sum of


number of terms (n) of the odd numbers 1, 3, 5, &c. beginning with 1, is equal to the square of that number (n).

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In the rule for summing an arithmetical series, let a = 1,b=2, and non, and the sum


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Q. E. D.

Theor. III. The difference of any two square numbers is equal to the sum of the two roots, together with twice the sum of the numbers in the natural scale between the two roots.

Let the one number be p, and the other Þ+, the intermediate numbers are pti, p+2, . ... &c. p+n-1. The difference of the squares of the given numbers is 2pn +12?; the sum of the two roots is 2p+n, and twice the sum of the series ptitp+2

&c. PtnI is (by Cor. 1. ist fect. of this chap.) 25=2p+nxnI, viz. the fum of the first and last multiplied by the number of terms, and it is plain that 2p+nt 2p+nxi-I=2pn+n. Therefore, &c.



Let r. be any number, and n any integer, r --1 is divisible by r-I. The quotient will be r

&c. till the index of r be o, and then the last term

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