exponent in the dividend and divisor, we shall find zero to be the exponent which it ought to have in the quotient; as divided by a3, for example, gives a°. To understand what is the import of such an expression, it is necessary to go back to its origin and to consider, that if we represent the quotient arising from the division of a quantity by itself, it ought to answer to unity, which expresses how many times any quantity is contained in itself. It follows from this, that the expression a° is a symbol equivalent to unity, and may consequently be represented by 1. We may then omit writing the letters which have zero for their exponent, since each of them signifies nothing but unity. Thus as be2 divided by a2 b c2, gives a1 bo co, which becomes a, as is very evident by suppressing the common factors of the dividend and divisor.

We see by this, that the proposition, every quantity which has zero for its exponent, is equal to 1, is nothing, properly speaking, but the explanation of a conclusion to which we are brought by the common manner of writing the powers of quantities by exponents.

In order that the division may be performed, it is necessary, 1. that the divisor should have no letter which is not found in the dividend; 2. that the exponent of any letter in the divisor should not exceed that of the same letter in the dividend; 3. that the coefficient of the divisor should exactly divide that of the dividend.

38. When these conditions do not exist, the division can only be indicated in the manner pointed out in the 2d article. Still we should endeavour to simplify the fraction by suppressing such factors, as are common to the dividend and divisor, if there are any such; for (Arith. 57) it is manifest, that the theory of arithmetical fractions rests upon principles which are independent of every particular value of their terms, and which would apply to fractions represented by letters, as well as to those which are represented by numbers.

According to these principles, we in the first place suppress the numerical factors common to the dividend and divisor, and then the letters which are common to the dividend and divisor, and which have the same exponent in each. When the exponent is not the same in each, we subtract the less from the greater, and affix the remainder, as the exponent to the letter, which is written only in that term of the fraction which has the highest exponent.

The following example will illustrate this rule.

Let 48 a b c d be divided by 64 as b3 ce; the quotient can only be indicated in the form of a fraction

48 a3 b5 c2 d

64a3 b3 c4 e

But the coefficients 48 and 64 being divisible by 16, by suppress. ing this common factor, the coefficient of the numerator becomes 3, and that of the denominator 4. The letter a having the same exponent 3 in the two terms of the fraction, it follows that a3 is a factor common to the dividend and divisor, and may consequently be suppressed.

To find the number of factors b common to the two terms of the fraction, we must divide the higher b' by the lower b3, according to the rule above given, and the quotient b2 shows, that b3 = b3 × b2. Suppressing then the common factor b3, there will remain in the numerator the factor b2.

With respect to the letter c, the higher factor being c4 of the denominator, if we divide it by c2 we shall decompose it into c2 x c2; and by suppressing the factor c2 common to the two terms, this letter disappears from the numerator, but will remain in the denominator with the exponent 2.

Finally, the letters d and e, will remain in their respective places, since in the state in which they are, they indicate no factor common to both.

By these several operations the proposed fraction is reduced to

and it is the most simple expression of the quotient, except we give numerical values to the letters, in which case it might be further reduced by cancelling the common factors as before.

39. It ought to be remarked, that if all the factors of the dividend enter into the divisor, which besides contains others peculiar to it, it is necessary after suppressing the former to put unity in the place of the dividend as the numerator of the fraction. In this case indeed we may suppress all the terms of the numerator, or in other words, divide the two terms of the fraction by the numerator; but this being divided by itself must give unity for the quotient, which becomes the new numerator.

Suppose for example the fraction

4 a2 b c

12 a2 b3 c

the factors 12, a2, b and c may be divided respectively by the factors 4, a2, b and c, or we may divide the two terms of the fraction by the numerator 4 a2 b c. Now the quantity 4 a2 b c divided by itself gives 1 for the quotient, and the quantity 12 a2 b3 cd divided by the first, gives by the above rules 3 b2 d; the new fraction then is

1

5 b2 d'

40. It follows from the rules of multiplication, that when a compound quantity is multiplied by a simple quantity, this last becomes a factor common to all the terms of the former. We may make use of this observation to simplify fractions of which the numerator and denominator are polynomials having factors that are common to all their terms.

by examining the quantity 6a3a2bc + 12 a2 c2, we see that the factor a2 is common to all the terms, since aa = a2 × a2, and that, besides, 6, 3 and 12 are all divisible by 3; so that,

6 as a2bc +12a2 c2=2u2 × 3a2bcx Sa2 + 4 c2 × Sa2. Also the denominator has for a common factor 3 a2; for the factors a2 and 3 enter into all the terms, and we have

9 a2 b― 15 a2 c +24 a3 = 3 b × 3 a2

-5c × 3 u2+8a × 3a2. Suppressing therefore the 3 a2 as often in the numerator as in the denominator, the proposed fraction will become

2abc + 4 c2

3b-5c+ 8 a

41. I pass now to the case where the numerator and denominator are both compound, and in which one cannot perceive at first whether the divisor is or is not a factor of the dividend.

As the divisor multiplied by the quotient must produce the dividend, it is necessary that this last should contain all the several products of each term of the divisor by each term of the quotient ; and if we could find the products arising from each particular term of the divisor; by dividing them by this term, which is known, we should obtain those of the quotient, after the same manner as in arithmetic we discover all the figures of the quotient by dividing successively by the divisor the numbers, which we regard as the several products of this divisor by the different fig

ures of the quotient. But in numbers the several products present themselves in order, beginning with the units at the last place on the left, on account of the subordination established between the units of each figure of the dividend according to the rank which they hold. But as this is not the case in algebra, we supply the want of such an arrangement by disposing all the terms of the dividend and divisor in the order of the exponents of the power of the same letter, beginning with the highest and proceeding from left to right, as may be seen with reference to the letter a in the quantities

5 a 22 a b + 12 a5 b2 — 6 a b3 — 4 a3 b1 +8 a2b3,

5 a* 2 a3 b+4a2 b2,

of which one is the product and the other the multiplicand in the example of art. 32. This is called arranging the proposed quan

tities.

When they are thus disposed, it is evident, that whatever be the factor by which it is necessary to multiply the second to obtain the first, the term 5 a7, with which this begins, results from the multiplication of 5 a4, with which the other begins, by the term in the factor sought, in which a has the highest exponent, and which takes the first place in this factor when the terms of it are arranged with reference to the letter a. By dividing then the simple quantity 5 a7 by the simple quantity 5 a*, the quotient a3 will be the first term of the factor sought. Now as the entire product ought by the rules of multiplication to contain the several particular products arising from the multiplication of the whole multiplicand by each term of the multiplier, it follows that the quantity here taken for the dividend, ought to contain the products of all the terms of the divisor, 5 a1- 2 a3 b + 4 a2 b2, by the first term of the quotient a3; and consequently, if we subtract from the dividend these products, which are 5 a7—2ao b+4a3 b2, the remainder — 20 a b + 8 a5 b2 — 6 a1 b3 4a3 b + 8 a2 b5 will contain only those, which result from the multiplication of the divisor by the second, third, &c. terms of the quotient.

The remainder then may be considered as a part of the dividend, and its first term, in which a has the highest exponent, cannot be obtained, otherwise than by the multiplication of the first term of the divisor by the second term of the quotient. But the first term of this part of the dividend having the sign, it is necessary to assign that which is to be prefixed to the corresponding term of the quotient. This is easily done by the first

rule art. 31, for the quantity — 20 a b, being regarded as a part of the product, having a sign contrary to that of the multiplicand 5 a*, it follows that the multiplier must have the sign. Division then being performed upon the simple quantities, — 20 a b and 5 a, gives-4 a b for the second term of the quotient. If now we multiply this by all the terms in the divisor, and subtract the product from the partial dividend, the remainder +10 at b3 —4 a3 ba + 8 a2 b3 will contain only the products of the third &c. terms of the quotient.

Regarding this remainder as a new dividend, its first term 10 a b3 must be the product of the first term of the divisor by the third of the quotient, and consequently this last is obtained by dividing the simple quantities, 10 a b3 and 5 at the one by the other. The quotient 263 being multiplied by the whole of the divisor furnishes products, the subtraction of which exhausting the remaining dividend, proves that the quotient has only three terms.

If the question had been such as to require a greater number of terms, they might evidently have been found like the preceding, and if, as we have supposed, the dividend has the divisor for a factor, the subtraction of the product of this divisor by the last term of the quotient ought always to exhaust the corresponding dividend.

42. To facilitate the practice of the above rules;

1. We dispose the dividend and divisor, as for the division of numbers, by arranging them with reference to some letter, that is, by writing the terms in the order of the exponents of this letter, beginning with the highest ;

2. We divide the first term of the dividend by the first term of the divisor, and write the result in the place of the quotient;

3. We multiply the whole divisor by the term of the quotient just found, subtract it from the dividend, and reduce similar terms.

4. We regard this remainder as a new dividend, the first term of which we divide by the first term of the divisor, and write the result as the second term of the quotient, and continue the operation till all the terms of the dividend are exhausted.

Recollecting that a product has the same sign as the multiplicand when the multiplier has the sign+, and that it has in the contrary case the sign (31), we infer that, when the term of the dividend and the first term of the divisor have the same sign, the