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Periodic continued fractions. The value of every periodic fraction may be determined by the solution of a quadratic equation; and the root of every quadratic equation may be expressed by a periodic fraction.

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To reduce the roots of a quadratic equation to continued fractions : let the equation be

am? +bx+c=0, (1) in which 62 - 4ac > 0.

The full point is placed over the first and last fraction of the period, as in circulating decimals, to denote the extent of the period.

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2 a

b? 4ac=1 4a.a,=b2 4a7.02

=&c. =d, suppose; then

6-dvd-b b=2a.e+b 2a,=

>e, and < e, +1; ;

2 a

63 - d Vd be=2a7.ey+b20,

>e, and <e,+1; 2a, 2a,

63-dvd-bg bz =2a2.e+ b2 | 2 az

>e, and < ez +1.

2a,
&c.
&c.

&c.
1 1
and x=et

ez tez tez + &c. In the same manner we may find the value of the other root of (1), viz.

1

(–6–62 4ac).

2az

za

The values of en, en, &c. are the same in this case, but they occur in an inverted order.

(G. A. 23; Lagr. Equ. Num. Ch. 6. Art. 2; Legendre, Theor. Nomb. 59—74.)

Ε

(31.) To find the value of Vc: the equation (1) and the above formula become respectively

– C=0,

1

a =

C-6 vc+b b=1.e-0

>e, and <e, +1; 1

a1

C-63 vc+ b2 b,=a7.e, -6, a,=

and <eg +1; a

a 2

c-b3|vc+bz bz =dz .ez az =

> ez and <ez +1.

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03

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1 1 1 and vo=e+

ei + e,+ ezt &c.

(B. 143_8; Legendre, Theor. Nomb. 28–33.) The periods in the series az, az, az, &c. @q, , ez, &c. for all values of c from 1 to 100 will be found in the annexed table. sa, a, az &c. 17{!

&c.

471 111110

Veerees &c. 18{I}

19{213128 20228

32
✓33{$38 1

5 25 31

34{11110

1 2 1 10 929 1 141 10

112118

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21 {5 4 3 4 51 22{93 23 6 1 /23{737)

J35{19 10 37{12 3831

39

1 2 4 2 18

32 31 11114

81

16 12

3 1 4.12

4 1 40 13 12

5 5 1 122 12 6 1

26{id

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27321

42{212

5 10
3431

31 26

28*3 2 3 10

43

1763 92 93 67 1
113151 311 12

13{433 41

74
44{85747581

1116

5 5 4 1

1 2 10
5 1
2 10

29{

30{
✓31{653 23 5 6 1

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1112111 12

5 4 1122 2 1 12

91

✓4594

1216
61
16

211 3 5 3 1 1 10

46{10376525673 10 1

1 311 2 6 2 113 1 12

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The last quotient in the period =2e, e being the nearest

(B. 149.) root.

GENERAL PROPERTIES OF EQUATIONS.

(32.) Every equation of n dimensions in x may be reduced to the form

xn-1 + an-2

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B

" ta

2001 – ? + +0,X ta=0; which for convenience may be represented by (@)=0.

If () be divided by x — a, the remainder will be (a).

If (@) N — a, a is a root of $ (a) = 0; and conversely. Every equation has at least one root.

(Du Bourguet, Ann. de Math. Tome 2.) Every equation of n dimensions has n roots, and no more. If is the goth root of Q(x)=0, then

Q(x)=(x – a) (x – a,)...... (x – ,). The coefficient of fom-1 in (x) is the sum of all the combinations of the quantities –a, –a,, - az, ... &c. – 0,19 taken n-m+1 at a time : • thus

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n+1 or 0 =

m

Sam

m

- 1; a, being unity: this will frequently be found to be a more convenient mode of arrangement than the former.

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