A Method of Approximating Towards the Roots of Cubic Equations Belonging to the
A Method of Approximating Towards the Roots of Cubic Equations Belonging to the
James Lookhart
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; fox- x^ -{-1=^ [6+2X x^ +cx-\-l) x^-\-l=^[lQ0x^-{-20x+l) a?^ + l = 10x4-l j;=10. It may be also solved accurately by means of the fractions, and by the method prescribed in Rule XIX. , but it is now to be done by approximation. — = 2352. 98, the cotemporary equation isj/^ — 2352. 98?/=2352, 98 v'2352. 98+. 4=48. 9+. Then, 2352. 98 . 75 2352. 23 . 020020=- 49. 9 2352. 250020, the root is 48. 500 adding . 5 49 as a second value of y Again, 2352. 23 2352. 25, the root is 48. 5 adding . 5 gives 4...9 as a third value of y. And here we must necessarily stop, because no new divisor can be obtained. It follows, that 49 is the exact value of y; and, taking ^=-r= TTir^l^j t^e accurate value of x is discovered, o 98 50 Ex. XV. Let ^3- 401=24 23_. L2=. 003 — = 111. 1, in all these equations^ the cotemporary equation is 3/^ — 111. 1^=111. 1 v^lll. 11 +. 4 = 10. 9 as a first value of y. Then, 111. 111111 . 75 110. 361111 1 . 084033 = 11. 9 110. 445144, the root is 10. 509 adding . 5 gives 11.
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