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XXXVIII. — Some Mathematical Researches. By H. Fox Talbot, Esq. 



(Bead 29th April 1867.) 



I. — On Cubic Equations. 



It is well known, that whenever the three roots of a cubic are all real, the 

 solution of the equation by Cardan's rule becomes illusory. This is the more 

 remarkable, because, a priori, one might have expected that the rule would only 

 fail when the roots were imaginary. Numerous researches have been made by 

 mathematicians on this subject ; but they have not succeeded in removing this 

 obstacle ; and the only mode of finding the roots of a cubic, when all three are 

 real, has been, by successive approximations, or the use of trigonometrical tables, 

 or (in the case of one root being a whole number), by tentative methods and trials 

 (which often succeed without much difficulty, when the coefficients of the equa- 

 tion are small numbers). 



I have found, however, that there exists a certain class of cubic equations 

 which can be solved by a process quite different from that of Cardan, and there- 

 fore not subject to any similar cause of failure. It is, moreover, exceedingly 

 direct and simple, requiring no extraction of the cube root. 



I shall suppose, for the sake of brevity, that the cubic equation wants its 

 second term. If otherwise, the second term must be taken away by the usual 

 rule. This being premised, the process which I speak of can be employed where- 

 ever the equation has a root of the form a + \/b, where a, b, are whole numbers 

 or rational fractions, and Vb is a Surd in its lowest terms. 



Since a + Vb is a root, it follows that a — Vb is also a root, and the third 

 root is — 2 a, since by hypothesis the second term of the equation is wanting. 

 All three roots are therefore real, and all the coefficients are either whole 

 numbers or rational fractions. 



Notation. — Let x n — px n ~ x + qx n ~* — &c. = be any equation. I denote 

 the coefficients in the usual way, by p, q, r, &c. One root of the equation will be 

 x, and I usually denote the other roots by y, z, &c. 



It is well known that p=x+y+s+ &c, q = xy + y z + xz + &c., 

 r = xy z + &c, and so forth. I adopt the abbreviated notation^ = S x, q = Sxy, 

 r = Sxys, and so on ; S standing for " the sum of," and Sxy meaning the sum 

 of all the combinations which can be made of two roots multiplied together ; 

 S xy z, of three roots, and so on. 



Most treatises on Algebra give easy rules by which to compute the values of 



VOL. XXIV. PART III. 7 Q 



