Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . , Let jy* — axy + x* zz o, be given. Put Ax for y, and the equation becomes A3*3* — a Ax l + x* zz o. Let n + i zz 3, and»~2j and the indices are 6, 3, 3 -, that is, 3, 6 — 3 zz 3, then r zz 3; and the leaft indicesbeing compared, the feries* will be an afcending one,which is this y zz Axz + Bx$ + O* + D*1 & fubftituted in the given equation will be asfollows : —
Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . , Let jy* — axy + x* zz o, be given. Put Ax for y, and the equation becomes A3*3* — a Ax l + x* zz o. Let n + i zz 3, and»~2j and the indices are 6, 3, 3 -, that is, 3, 6 — 3 zz 3, then r zz 3; and the leaft indicesbeing compared, the feries* will be an afcending one,which is this y zz Axz + Bx$ + O* + D*1 & fubftituted in the given equation will be asfollows : — axy + x* — a Ax —aBx6— aCx? — aDxx*+ I*3 Then aA zz 1, and A zz i-, B zz i, C zz Jj a* a* of Dzzii, &c. Whencey-t + *1 + $£al° J a T a\ a\ + &C. Ex. 6. Let y* — byz + gbxx — x* zz o. Subftitute Axn for y and the equation is A5 gvx — *3 Here £A2 zz 9J; and A zz 3 : alfo B zz — Whence RULE. If the equation determining A, be an adfecledequation, which has feveral equal roots or valuesof A, then you muft divide the lead remainder,found by Rule 1, by the number of equal roots,one of which you take for A ; and take this quo-tient for another remainder. Or die divide rfound by Rule 2, by that number, and make ufeof the quotient, inftead of r. Ex. 7. Let y> — xy* + 2x1y1 — x>y — *,4 zz o, to Jlnd y. Put Ax for y9 and the equation becomes A9x> — AV + 2Azx — Ax TJ—*,+— o. Let 3« + i zz in + 2 ; whence n zz 1, then the in-dices are 9, 4, 4, 4, 14. But the fum of thecoefficients for the lead index 4, is —A -J- 2AX— A zz o, or A* — 2 A + 1 zz o, which equation has two r92 INFINITE SERIES. B. I. two equal roots A zz r, and A zz i. Now thedifference of the indexes will be 5, 10 5 there-fore divide 5 by 2, gives —, and we have — 2 7. y 5, 10 for the differences. Therefore r, s, t9 &c. will be —, 5, 7—, 10, &V. Or (Rule 2) r zz
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