10. Add the several results furnished by each of the questions in Case III., taking each of the examples in that case as an example in this.

To subtract one fractional quantity from another.

REDUCE the fractions to a common denominator, (the least is the more elegant) if they have not a common denominator.

Subtract the numerators from each other, and under their difference set the common denominator.

3a

their difference? Likewise when taken with sign of a changed, viz.

+ 6 and

4

4a

7

To multiply fractional quantities together.

MULTIPLY the numerators together for a new numerator, and the denominators for a new denominator *.

1. When the numerator of one fraction, and the denominator of the other, can be divided by some quantity, which is common to both, the quotients may be used instead of them: or, in other words, the fractions may be reduced to their lowest terms before they are multiplied together.

2. When a fraction is to be multiplied by an integer, the product is found either by multiplying the numerator, or dividing the denominator by it; and if the integer be the same with the denominator, the numerator may be taken for the product. This may be readily deduced as a case of the general rule.

To divide one fractional quantity by another.

DIVIDE the numerators by each other, and the denominators by each other, if they will exactly divide: but, if not, then invert the terms of the divisor, and multiply by it as directed in multiplication, p. 143 *.

* If the fractions to be divided have a common denominator, take the numerator of the dividend for a new numerator, and the numerator of the divisor for the new denominator. See 3. below.

2. When a fraction is to be divided by any quantity, the value is the same whether the numerator be divided by it, or the denominator multiplied by it.

3. When the two numerators, or the two denominators, can be divided by some common quantity, let that be done, and the quotients used instead of the fractions first proposed. This is obvious from the circumstance that if these quantities be suffered to remain, they constitute a common factor, or a common measure of the quantity which results from the division.

10. Suppose we divide a3 + y3 by x3 — y3, and then divide the result by the quotient of x + y by x ży?

=

-

y; what final quantity shall we get, provided x

11. If we add three-fourths of a number to one-half its square, and divide the sum by three-eighths of its cube; and moreover, if we subtract three-fourths of that number from half its square, and divide the result by minus one-eighth of the square; what is the quotient of the former by the latter result?

CASE X.

Continued fractions.

....

Ir we recur to the process for finding the greatest common measure, p. 133; and denote for simplicity the functions X, X, X, .......... by a, b, c ..., and the quotients Q, Q,, Q2 .. by a, ß, y ...; and if we also denote the divisions on the left by fractions; then we shall have those two columns converted into the following:

:

The expression (either form) on the left is called a continued fraction. It has many curious properties and in several inquiries is of great value. The only

* The idea of continued fractions was first started by Lord Brouncker, the first President of the Royal Society; but the method owes its present elegant form to Lagrange, who made ex

one to which, however, the student of this course will have occasion to apply it, is to find a series of fractions converging towards the true value of the given fraction, but having its terms expressed in smaller numbers than a and b.

The formation of the values of the terms of the converging fractions has been stated at p. 134, art. 4, where the upper line expresses the numerator, and the lower one the denominator of the converging fraction at the first, second, third, and successive steps.

EXAMPLES.

in the form of a continued fraction, and find the con

And forming the converging fractions according to the rule, they are

The properties of these fractions are:

1. That they are alternately less and greater than the true value.

2. That each of them is in its lowest terms.

3. That each of them is a nearer value than any other that can be formed without taking higher values of the numerator and denominator.

4. That if and be two consecutive converging fractions, then

Pm
Im

Pm+1
qm+1

Pm 9m+1 Pm+1 9m = 1. See Hind's Algebra, pp. 284-306.

tensive use of it, both in the solution of algebraical equations with numeral coefficients, and the solution of indeterminate equations.

The latest improvements in the use of the method as applied to the solution of algebraical equations, were made by the late Mr. Horner, and published in the Annals of Philosophy and Quarterly Journal. The second notation above given was proposed by Sir John Herschel, and for economy of space, both in writing and printing, is the most convenient. For a good view of the subject in an elementary form, the reader who wishes to go further into the subject may consult Hind's Algebra or Young's Equations :-both of them works deserving of cordial recommendation.