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 ax2 + bx+c=0, (1) in which b2-4ac >0. a 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. b2-4acb-4a.a,b-4a,.a2 = &c. In the same manner we may find the value of the other root of (1), viz. The values of e1, e, &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.) E (31.) To find the value of c: the equation (1) and the above formulæ become respectively (B. 143—8; Legendre, Theor. Nomb. 28—33.) The periods in the series a1, α, 3, &c. е1, C2, C3, &c. for all values of c from 1 to 100 will be found in the annexed table. GENERAL PROPERTIES OF EQUATIONS. (32.) Every equation of n dimensions in a may be reduced to the form which for convenience may be represented by (x)=0. If (a) be divided by x-a, the remainder will be (a). ß If (x) x − ɑ, versely. a is a root of p(x) = 0; and con 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 a, is the 7th root of (x)=0, then (x) = (x − a)(x — α)...... (x — ɑ„). ' The coefficient of am-1 in combinations of the quantities (x) is the sum of all the a1, -α, —α, ... &c. — an -- a, being unity: this will frequently be found to be a more convenient mode of arrangement than the former. |