hence A10, or A= 1; B-a A3= 0, or B = a;

C-3a AB+b=0; or C = 3a2-b; &c.

therefore y = x + a x3 + (3a2 − b) x3 + &c.

The method of determining the proper series to be assumed in each case, without previous trial, is given by Maclaurin, Alg. Part 11. Chap. 10.

CONTINUED FRACTIONS.

a

325. To represent

in a continued fraction.

b

b) a (p

c) b (g

d) c (r

e &c.

Let b be contained p times in a, with a remainder c; again, let c be contained q times in b, with a remainder d, and so on; then we have

326. COR. 1. An approximation may thus be made to the value of a fraction whose numerator and denominator are in too high terms; and the farther the division is continued, the nearer will the approximation be to the true value.

327. COR. 2. This approximation is alternately less and greater than the true value. Thus p is less than is less than; and

a

is greater, because a part of the denominator of the

1

fraction is omitted: 9+ is too great for the denominator,

reduced to simple fractions, are called the "Converging Frac

Ex. To find a fraction which shall be nearly equal to

3+

The first approximation is 3, which is too little, the next is

too little; and so on.

cumference of a circle whose diameter is 1; therefore the cir

This fraction expresses, nearly, the cir

converging fractions.

Let the continued fraction be (Art. 325)

P pq+1 (pq+1) r+p (pq+1.r+p) s + pq + 1

p

or

in which the law of formation is observed to be as follows:

&c.

Write down in one line the quotients p, q, r, s, &c., and the first and second fractions at sight, then the other fractions may be obtained thus:

For the 3rd.

num".

=

3d quot. num". of 2d fract. + num". of 1st fract. denom'. 3d quot. x denom". of 2d fract. + denom'. of 1st fract.

=

For the 4th.

'num'.

=

4th quot. num". of 3d fract. + num". of 2d fract. denom". 4th quot. x denom". of 3d fract. + denom". of 2d fract.

=

n-1

And generally, for the nth fraction in the series, Multiply the nth quotient by the numerator of the fraction and add the product to the numerator of the n fraction. This will give the numerator.

Multiply the nth quotient by the denominator of the n-1 fraction, and add the product to the denominator of the n-2 fraction. This will give the denominator.

Ex. To find a series of converging fractions for

84

227

th

th

th

th

329. To express a + 1 in the form of a "Continued

Fraction."

&c. each of which is nearer to the true value

of 17 than the one preceding.

330. To express 11 in a continued fraction; and to find the converging fractions.