An Essay On the Resolution of Algebraic Equations

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34- I3. 14)- (II . 23"- 12 . R3) (Tl. 24-T2. 14)}/2-A; -h{(ri. 22-1^. 12)(n. 4r-U. 14)~(Ii. 24-12. 14)2}/t2 = 0.
THE QUINTIC. 79 This can be divided into two parts by means of the dispos- able constants ; the first two terms being made equal to zero ; and the last three terms (which contain only h and k) being also separately equated to zero. We thus have a quadratic in h and k with rational coefficients, (which I propose to call the first quadratic), whose solution gives a linear relation be-
...tween h and k. The coefficient of this relation is a quadratic- irrational of the 30th degree.
Substituting for h in the equation formed by the sum of the first two terms above being made zero, we have another quadratic equation with quadratic-irrational coefficients, (which I call the second quadratic) ; and its solution gives a linear relation between m, I and k, whose coefficient is what we may call a quadratic-quadratic-irrational. We have thus linear values for k and h in terms of I and m. Sub- stituting these in the cubic equation of condition, which, in this notation, is .


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