x-sx"+tx-5+vx-6+ &c.=0, thats is greater than the greatest root; and in the equation of three terms x^-vx+w=0, that "v is greater than the greatest root of the equation*. 35. There are several methods also of finding quantities which shall be less than the greatest roots of equations, of which the following are examples. I. In the equation (A)=0, whose roots are a, ß, y, d, &c., we have (Art. 2) q=aß+ay+ad+ßy+ßd+yd+&c., of which the number of quantities (by Art. 143, El.) is n("~1'); if therefore « be the greatest root, n("-")«° a be greater than q, for each of the quantities aß, is less than 2; hence a must be greater than must II. If the roots of the equation (A)=0 be a, ß, — 1, -, &c., then (by Art. 26) a2+ß2+ y2+d2+ &c. (to n quantities) = p2 - 2q+; if, therefore, a be greater than any of the other roots, na' will be greater than p'~ For these equations may be written thus; xn− rxn−3+sx »4+tx»-5+&c.= (x 3 — r) xn−3+s xn−s+txn−5+&c.=0. · x" — s x "→→tæn−3+vxn-6+ &c. = (x 4 — 8) xn−4+¿xn−5+vxn−6+&c.=0 x" — v x+w = (x."——v)x+w=0; substituter and r+k; s and s+k; " and "+k, for x, in these equations respectively, and the truth of the rule will appear. + For a2+62+22+2+&c. is the coefficient of the second term of the equation whose roots are the squares of the roots of the equation (A)=0; which coefficient when referred to the Table in Art. 2 (since a2, 62, y2, are positive) must have a negative value, i. e. it must be -(2g-p2) or p2-2q. p'-2q, and consequently greater than p-29, n 29 less than a. n or If, on the other hand, be greater than any of the other roots, then ' will be greater than po-29, and consequently -√√√P n quantity less than -d. 2 a negative n III. Supposing the roots of the equation (A)=0 to be the same as before, then «”ß*+a3y® +ß3y®+y*d2+&c. (to n("—1) 2 2 2 quantities) q−2pr+2s*; if, therefore, be greater than any of the other roots, n("=1) must be greater than q'-2pr+2s, and consequently a greater than 292-4pr+48 less than a. But n(n-1) or if be greater than any of the other roots, then 2q2-4 pr+4s will be greater than n(n-1) and consequently - 4 29-4p+4s will be a negative quantity less n(n-1) than -d.t 2 • For a2 ß2+a2 z2 + ß2 z2 + z 2 d2+&c. is the coefficient of the third term of the equation whose roots are the squares of the roots of the equation (A)=0. (Art. 26.) α + With respect to all these rules, it may be observed, that "if a be much greater than any of the other roots (or indeed if all the roots are “not pretty nearly equal to each other) the quantities (), will be much greater than q, p2-2q, q2—2 pr+2s, "so much less than «, as to be no approximation to that root." SECT. II. On the method of finding a limiting equation to any given equation. 36. The roots of the equation (A)=x”—px" ̄'+qx"¬¬rx"¬3+&c...±tx2+vx±w=0, arranged according to the order of their respective magnitudes, being a, ß, -, -, &c., let this equation be transformed (by Art. 21) into one whose roots shall be ɑ— —a, ß—a, —~—ɑ, -d-a, &c., and let the resulting equation be y"+Py+Qy2+ &c. ... + Sy3+Ty®+Vy+W=0, then (the coefficient of the last term but one of this equation) is equal to na"~' — (n − 1) par2+(n−2) qan-s-&c....±2ta‡v. But (by Art. 2) the coefficient of the last term but one of an equation of n dimensions is equal to the sum of the products of any n-1 roots; hence, V=(a—α) (a—ß) (a + y) &c. +(a−a) (a−ß) (a + d) &c. Now let the roots a, ß, −1, −d, &c. of the equation (A)=0 be successively substituted for a in this value of V, then (since three of these combinations of roots become equal to 0, at each step of the operation) the results will stand thus; When Since the coefficients of the equation y"+Py"—1+Qy”¬2+Ry +"~3&c.=0 are all assumed positive, this equation must be equal to (y+a—a) (y + ß—a) (y—y—a) (y—d—a) &c.=0 =(y-(aa)) (y-(ap)) (y-(a+z)) (y = (a+d)) &c.=0; in finding the value, therefore, of the coefficients of this equation with reference to the table in Art. 2, the quantities to be combined are ɑ−x, a—ß, a + y, a +d, &c. When a is substituted for a, V= (a-B) (a+y) (a+d) &c. =+ h for a, V= (8-α) (3+2) (3+ d) &c.——i for a, V=(-a) (— y — ß) ( − y + d) &c.=+k for a, V=(—d—a)(— d—ß)(−d+2) &c.=−/ V= &c. &c. &c. =+&c. where the signs of the quantities h, i, k, l, &c. are deduced from the relative values of a, ß, −y, −d, &c. as in Art. 32. 37. From hence it appears, that when the roots a, ß, -,-, &c. of the equation (A)=0 are substituted successively for a in the equation V=0, or for x in the equation nxTM — (n−1)px2 + (n−2 ) q x3-&c....+2txv=0, the results are alternately + and -; the n roots of the equation (A)=0 are therefore (by Art. 32) limits between the (n-1) roots of the equation - nxTM-1 — (n − 1) p x22 + (n−2) q x3-&c.... ± 2tx=v=0; and vice versa, the (n-1) roots of this latter equation are limits between the n roots of the equation (A)=0. Let these n-1 roots be 7, 1, -, &c. and the arrangement is nx^~^ — (n − 1)px"→2 + (n−2) 9x^~3 —&c... ±2txv=0 q is called the limiting equation to the equation (A)=0; and it is derived from (A)=0" by multiplying each term of that "equation by the index of x, and then diminishing the index "of that term by unity, in the same manner as V is derived 'from Win Art. 22. H 38. The 38. The limiting equation to the equation (V)=0, found by this rule, is, n(n − 1)x”—2 — (n − 1)(n−2) px2¬3+(n − 2) (n − 3) q xTM~—&c...±2t=0, for it is derived from (V)=0 in the same manner as (V)=0 is derived from (A)=0. Divide this equation by 2, and it becomes which (by Art. 21) is the coefficient (T)=0 of the transformed The limiting equation to 6x+27x+18=0 is 4x+9=0, in which x=-=-2.25. The series of limiting equations to the equation x1+9x3+9x2-41x-42=0, with their corresponding roots, may therefore be thus arranged; Roots |