## An essay on the resolution of equations |

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0Qo+lQ Accordingly algebraical resolution applicable arise become biquadratic circular substitution coefficients coincide collapse conducted consider cubic equation deduce denote derived Descartes designate determinate non-evanescent function different values effect elements eliminate equa equal to zero equation aa equation in x equations of condition evanescent expression fifth degree final equation fourth degree functions of Q Hence homogeneous equations indeterminate indices integer integral functions interchanges involved Lagrange and Vandermonde manifest manifestly mth degree non-symmetrical function number of different observing obtain order of succession original equation perceive proposed equation quadratic equation quantities rational and integral rational functions reducible remains fixed represent requisite substitutions resolution of equations result roots satisfy sets of values simultaneous equations solution solving successively equal suppose susceptible symbol symmetrical functions theorem third degree tions Tschirnhausen vanish Whence xv xv

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Page 82 - ... or, in other words, to express a root of the general equation of the fifth degree by a finite combination of radicals and rational functions. The problem had long engaged the attention of mathematicians, and was attacked by the most celebrated analysts of the last century with great skill and vigour, but without success. In the early part of the present century, Abel, the young and gifted Norwegian mathematician, attempted...

Page 49 - And if #„ xt, x^, xv xb are the roots of the given equation of the fifth order, and it' «, /3, «y, 8, e represent in an undetermined or arbitrary order of succession the five indices 1, 2, 3, 4, 5, and if t denote an imaginary fifth root of unity (I conform myself to Mr. Jerrard's notation), then P\> Pv 1}3> and the other auxiliary quantities /, u, are obtained from the system of equations — * Communicated by the Author.

Page 83 - Let us for a moment consider the nature of the difficulty which had to be overcome.