Extension of the Celebrated Theorem of C Sturm Whereby the Roots of Numeral Eq

James Lockhart

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Let * 3 — a* 2 + c = — — separates the roots nearest to each other.
Thus let * 3 - 7* 2 + 36 = where * = — 2, + 3, + 6 324 the separator is — — = 3*3 ... .
and let x s — 9* 2 + 10 = where * = — 1, 5 +_ ^15 the separator is -— = -55 between the negative and least positive root.
162 In all the preceding cubic equations if the roots are in arithmetical pro- gression the middle one is obtained, or if two of them are equal, one even root is derived.
Let x* + ox 3 + 1 * 2 - 70* + 120 = where * = 2, 3... (b 2 + \2d)c . .. .
separates the two positive roots 9c 2 + 2b 3 - 8bd This gives = 2*3 .... Nearest to the least root.
10 EXTENSION OF THE Let x* + ox 3 — bx 2 + ex — d — Ohave three positive roots, the function "2 b y 2 — Scy + 4d = contains two separations.
and — -^ — ~ ' separates two roots, or exhibits equal roots.
Let x n — ax 2 + b x — c = 0, have three positive roots, n being a whole number, greater than 2, the function n — 2 a x 2

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