502. When m and n are nearly equal to each other, and much less than r, s, &c. and also both positive or both negative, then which is an approximation to m the less of the two; but if one 1 1 m n of these quantities be positive and the other negative, + may be either positive or negative, and greater, equal to, or 1 1 P less than,- + - + &c. and consequently is not necessarily an Q approximation to any of the quantities m, n, r, s, &c. Let P - Qv + Rv2= 0; the roots of this equation will be m and n, nearly. For if m, n, r, s, be the roots of the equa tion P P − Qv + R v2 — Sv3 + v1 = 0, = mnrs, Q =mns + mnr + mrs + nrs, R = mn + mr + ms + nr + ns + rs, and since m and n are small when compared with r and s, Q = mrs + nrs nearly, and R = rs nearly; therefore the equation P – Qv + Rv2 = 0 becomes whose roots are m and n. By the solution then of this quadratic a much nearer approximation is made to the root a + m than by the former method; and at the same time an approximation is also made to the root a + n. 503. In the same manner, if t roots be nearly equal, in order to approximate to them, it will be necessary to solve an equation of t dimensions. See Waring's Med. Algeb. p. 186. 504. If we have two equations, containing two unknown quantities, we may discover the values of these quantities nearly in the same manner. Find, by trial, approximate values of x and y; such are 20 and 1; and let ∞ = 20v, y = 1 + x; then y = 400 + 40v + 400≈ + v2 + 40vx - v2x = 405, and xy — y2 – 19 → v + 18≈ ÷ v≈ = 20, = and neglecting those terms in which ≈ or v is of more than one dimension, or in which their product is found, as being small when compared with the rest, By making use of the values thus obtained nearer approximations may be made to x and y. [505. To approximate to the roots of an equation by Continued Fractions. (Lagrange's Method.) 2-1 First Let the proposed equation be a"-px"1+ &c. = 0. discover, by the method of Art. 462, the integral limits of its possible roots, let one of them lie between a and a+1. Assume 1 and let y" - p'y-1+ &c. = 0 be the transformed 1 equation; and since - lies between 0 and 1, y has one possible value greater than 1. y Suppose ẞ found, as before, the nearest 1 integer less than this value, and assume y = ß+ and let -1 "-p"-1 + &c. = 0 be the transformed equation, which, as before, must have one possible root at least greater than 1. and thus an approximation is made to the value of a to any required degree of accuracy. Ex. To find one of the roots of the equation ∞3−2x-5=0 in the form of a continued fraction. Comparing this equation with a3-qa+r = 0, we find that 93 25 8 4 27 4 27 = a positive quantity; therefore the equation has two impossible roots; and the third root is positive, since the last term is negative. Substituting 2 and 3 for æ, the results are 1, and +16; therefore the root lies between in which 10 and 11 being substituted, the results are – 61, and the value of which may be found in a series of convergents.] SUMS OF THE POWERS OF THE ROOTS 506. Let a, b, c, &c. be the roots of the equation and S1, S2, S3, mth Sm, the sum of the 1st, 2nd, 3rd,...... ', powers of the roots respectively; then will S1 = p, S2 = pS, -2q, S2 = p S2-qS1 + 3r, &c. S3 and generally Smp Sm-1qSm-2 + rSm-3...... mw; where+w is the coefficient of the m+1th term. whatever be the value of x; and, by actual division, and if x be supposed greater than any of the magnitudes a, b, c, &c. no quantity is lost in the division; therefore, by addition, <-1 2-2 n-3 nx11- (n − 1) px2-2 + (n − 2) q x2¬3... + (n −m) wx2-m−1 _ &c. and equating the coefficients of like powers of x, (Art. 324). Sm-pSm-1+qSm-2-rSm-3...... + nw = (n − m) w, Ex. Let mw. +5x2-6x-8=0; then by comparing the terms of this with the terms of the equation Sum of the squares (S2) = pS1 − 2q = 25 + 12 = 37. H Sum of the cubes (S3) = pS2 – qS1 + 3r = 185 30+ 24 = - 191. 507. The proposition also admits of the following proof. I. The same notation being retained, let m and n be equal, and since a, b, c, &c. are roots of the equation, |