and a±x)a2+x2(a±x +

2x2 a±x

Answer as required.

a tax

2x2

Fax+ Here at is the integral part of the quantity.
±ax-x2 2x2
and is the fractional part.

Remainder 2x2 a±x

a±x+ is the mixed quantity required.

Therefore

CASE III. Fractions reduced to a common denominator. RULE. Multiply each numerator separately by every denominator except its own, for the new numerator, and all the denominators together for a common denominator.

each pair to equivalent fractions, that shall have common denomi

3xX2X(a-x)=6ax-6x

new numerators.

(a+x)X2X7=14a+14x

2X7X(a-x)=-14a-14x common denominator.

a+x

Hence the fractions required are adf cbf ebd and

bxdxf=bdf=common denominator. bdƒ' bdf'

5. Reduce 2+* and

bdf

mon denominator.
(a+x)X(a+x)=a2+2ax+x2 }
(a—x)X(a—x)—a2—2ax+x2)
(ax)X(a+x)—a2—x2

The common denominator.
Here 5x×3×2x=30x2
(a-x)X(a+x)2x-2ax-2x3
1X(a+x)X3-3a+3x

(a+x)X3X2x=6x+6x

6. Reduce

a+x' 3

new numerators.

} Ans. 2+2a2+2 and

a 2y aty
and
3'7' a-y

mon denominator.

Here xx5X(1+x)=5x2+5x

(x+1)×3×(1+x)=3x2+6x+3 (1-x)X3X5-15-15x

3X5X(1+x)=15+15x common denominator.

axbX(a—y)—a baby 2yX3X(a-y)=6ay-by (a+y)x3xb=3ab3by 3xbx (ay)=3ab-3by

common denominator.

a2+2ax+x2
a2-x2

-

; or 3' 5

12+1

and

1. -X

to a com

each set to a common denominator.

First (b+c) (ab) —ab+ac—b2—bc
(a—4e) (a+b)—a2+ab-4ae4be

}

new numerators.

(a+b) (a—b): a2b2 the common denominator.
ab+ac-b2-bc a2+ab-4ae-4be

Hence, the factors are

and

a2-b2

8a1+4a2x-2ax2
2aX4a2

a2 8x1-4ax2+2a2x

2xX4x2 4a3x2+2ax3-x1 4a2x2

On the Method of Finding the Greatest Common Measure of two

or more Quantities.

43. One quantity is said to measure another, when it is contained in that other a certain number of times, without a remainder. 44. A quantity is said to be a multiple of another, when it contains that other quantity a certain number of times, without a remainder.

45. A common measure of two or more quantities is any quantity which measures them all; and the greatest common measure is the greatest quantity which will so measure them. Thus, 2a is a common measure of the quantities 24ab2, 16a2bc, and 12abc2, and their greatest common measure is 4ab.

46. If one quantity measures another, it will also measure any multiple of that quantity. Thus, let b measure a by the units in m, then amb; and let na be a multiple (denoted by the units in ?) of a, then na―nmb; consequently b measures na by the units

in nm.

47. If one quantity measures two others, it will also measure their sum and difference. For let c measure a by the units in m, and b by the units in n, then a=mc, and b―nc; therefore, a+b) =mc±nc=(mn)c; consequently c measures a+b (their sum) by the units in m+n, and a-b (their difference) by the units in m-n. (The quantity ab means a plus or minus b.)

48. The Rule for finding the greatest common measure of two numbers may be thus investigated. Let a and b be any two numbers, whereof a is the greater; and let the following operation be performed upon them: viz.

b)alp

pb

c)b(q

qc

d)cir

rd

Where a divided by b gives the quotient p, and remainder c; b divided by c, the quotient q, and remainder d; c divided by d, the quotient r, and remainder 0. Then, since in each case the dividend is equal to the divisor multiplied by the quotient plus the remainder, we have c=rd.

b-qc+d(since qc=qrd) qrd+d=(gr+1)d

a=pb+c= {and since pb=(pqr+p)d} (pqr+p+r)d. Hence, since

P, q, r are whole numbers, d is contained in b as many times as there are units in gr+1, and in a as many times as there are units in pqr+p+r; consequently the last divisor d is a common measure of a and b; and this is evidently the case, whatever be the length of the operation, provided that it be carried on till the remainder is nothing.

The last divisor d is also the greatest common measure of a and b. For let x be any common measure of a and b, such that a=mx, and b―nx, then

=a—pb—mx—pnx—(m—pn)x d—b—qc=nx—(qm—pqn)x—(n—qm+pqn)x; .. x measures d by the units in n―qm+pqn, that is, every common measure of a and b measures d. Now it has been shown that d is a common measure of a and b; and the greatest measure of d is evidently itself; consequently d is the greatest common measure of a and ỗ. Hence this rule for finding the greatest common measure of two numbers: "Divide the greater by the lesser, and the preceding divisor by the last remainder, till nothing remains; the last divisor is the greatest common measure.”

To find the greatest common measure of three numbers, a, b, c; let d be the greatest common measure of a and b, and x the greatest common measure of d and c; then x is the greatest common measure of a, b, and c. For, let a=md, b-nd, dpx; then a―mpx, and b=npx, therefore x is a common measure of a and b; and, since it also measures c, it will be a common measure of a, b, and c. But, as above, every common measure of a and

measures d; therefore, every common measure of a, b, and c, measures d and c; and consequently the greatest common measure of d and c, or x, will also be the greatest common measure of a, b, and c.

In general, let there be any set of numbers, a, b, c, d, e, &c.; and let x be the greatest common measure of a and b ; y the greatest common measure of x and c; z the greatest common measure of y and d; &c., &c.; then will y be the greatest common measure of a, b, c; z the greatest common measure of a, b, c, d; &c., &c.

49. To find the greatest simple common measure of Algebraic quantities, the rule is, "to find the greatest common measure of their coefficients, and then annex to it the letters common to all the quantities;" thus the greatest common measure of 24ax'y', 16bxy, and 6axy, is 2xy.

To find the greatest compound common measure of two alge