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. 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. 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 – bę 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. 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 – bę az = > ez and <ez +1. 03 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 5 25 31 34{11110 1 2 1 10 929 1 141 10 112118 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 27321 42{212 5 10 31 26 28*3 2 3 10 43 1763 92 93 67 1 13{433 41 74 1116 5 5 4 1 1 2 10 29{ 30{ 1112111 12 5 4 1122 2 1 12 91 ✓4594 1216 211 3 5 3 1 1 10 46{10376525673 10 1 1 311 2 6 2 113 1 12 GENERAL PROPERTIES OF EQUATIONS. (32.) Every equation of n dimensions in x may be reduced to the form xn-1 + an-2 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 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. |