Thence the general rule for multiplying two simple quantities having any exponents whatever, is as follows; we must add together the two exponents of the same letter; this is the same with that established in article 16, for quantities having entire exponents. We shall find, according to this rule,

Division. In order to divide the one by the other two simple quantities having any exponents whatever, we must follow the rule for quantities having entire and positive exponents, that is, for each letter we must subtract the exponent of the divisor from that of the dividend.

For, the exponent of each letter in the quotient must be such that when added to that of the same letter in the divisor, the sum shall be equal to the exponent of the dividend; then the exponent of the quotient is equal to the difference between the exponent of the dividend and that of the divisor.

We shall find, according to this rule,

-?) ; = a3—(−1) = a11;

3

a: a

4

Formation of Powers. In order to raise a simple quantity having any exponent whatever to the mth power, conformably to the rule given in article 167, we must multiply the exponent of each letter by the exponent m of the power, since raising the quantity to the mth power, is to multiply it m1 times by itself; then, according to the rule for multiplication, we must add the exponent of each letter m 1 times to itself, or multiply it by m.

Extraction of Roots. In order to extract the nth root of a simple quantity, conformably to the rule in article 167, we must divide the exponent of each letter by the index n of the root.

For, the exponent of each letter in the result must be such, that, when multiplied by the index n of the root to be extracted, the product shall be the exponent which the letter has in the proposed simple quantity; therefore, the exponents in the result must be respectively equal to the quotients arising from the division of the exponents in the proposed simple quantity, by the index n of the

The last three rules have been easily deduced from the rule relative to multiplication; but they might be demonstrated by going back to the original quantities having any exponents whatever. We conclude, by an operation equivalent to a demonstration, which embraces implicitly the two preceding.

For, if we go back to the origin of these notations, we find that

The advantage derived from the employment of exponents, of any nature whatever, consists principally in this, that the calculation of expressions of this kind requires no other rules than those which have been established for the calculation of quantities having entire exponents. Moreover these calculations are reduced to simple operations upon fractions, operations with which we are very familiar.

Of Progression by Differences and by Quotients.

Progression by Differences.

181. We give the name of equidifference, arithmetical progression, or progression by differences to a series of terms, each of which exceeds or is exceeded by that which precedes it, by a constant quantity which is termed the ratio or difference of the progression. Thus, let there be the two series

1, 4, 7, 10, 13, 16, 19, 22, 25...

60, 56, 52, 48, 44, 40, 36, 32, 28...

The first is called an increasing progression, of which the ratio is 3, and the second a decreasing progression, of which the ratio

is 4.

Let us designate, in general, by a, b, c, d, e, f... the terms of a progression by differences; it is usually expressed thus ;

÷a. b C. d e.f.g .h.i.k...

and we read it in this manner,

a is to b, as b is to c, as c is to d, as d is to e...,

or, more concisely,

a is to b is to c is to d is to e...

This is a series of continued equidifferences, in which each term is at the same time both consequent and antecedent, with the exception of the first term, which is only an antecedent, and the last, which is only a consequent.

182. Let us call r the ratio of the progression, which we will suppose increasing, in all which is to follow. (If it were decreasing, it would be sufficient to change r tor in the results.)

This being laid down, we have evidently, according to the definition of the progression,

b=a+r, c=b+r=a+2r, d=c+r=a+3r...; and, in general, a term of any place whatever is equal to the first, plus as many times the ratio as there are terms before that under consideration. Thus, let 7 be this term, and n the whole number of terms to this, inclusive; we have for the expression of this general term,

1 = a + (n − 1) r.

For, if we suppose n = 1, 2, 3, 4, . . . successively, we shall find the first, second, third, . . . term of the progression.

If the progression were decreasing, we should have, on the other hand,

The formula, = a + (n

1) r, serves to give the expression of a term of any place whatever, without being obliged to

determine all those which precede it.

Thus, if we seek the 50th term of the progression

÷ 1.4.7.10. 13. 16. 19 ...

we have, by making n = 50,

1 = 1 + 49.3 = 148.

183. A progression by differences being given, it may be proposed to determine the sum of a certain number of terms. Let there be the progression

a.b.c.d.e.f...i.k.l,

continued to the term 7 inclusive, and let n be the number of terms, and r the ratio.

We begin by observing, that if a designates a term which has terms before it, and y a term which has p terms after it, we have, according to what has just been said, the equalities

which shows, that in every progression, the sum of any two terms, taken at equal distances from the extremes, is equal to the sum of the extremes; or the two extremes, and two terms taken at equal distances from these extremes, form an equidifference (in the order in which they are written.)

This being admitted, let us suppose that we have written the progression under itself, but in an inverse order, thus

Let us call S the sum of the terms of the first progression, 2 S will be the sum of the terms of the two progressions; and we shall have, by uniting the terms in the same vertical column,

28=(a+1)+(b+k)+(c+i)+...+(i+c)+(k+b)+(l+a);

or indeed, since all the parts a +1, b + k, c+i..., are equal, and in number n.

2 S = n; S=

(a + 1) » ; then finally $ = (a + 1) n;

2

that is, the sum of the terms of a progression by difference is equal to the product of the sum of the extremes multiplied by half the number of terms.

If, in this formula, we substitute for l its value a + (n we obtain further

1) r,

but the first expression is most used.

Applications. We require the sum of the first fifty terms of the