On the Equation of the Squares of the Differences of a BiquadraticAuthor(s): John CaseySource: Proceedings of the Royal Irish Academy. Science, Vol. 2 (1875 - 1877), pp. 40-41Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489982 .

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40 Proceedinys of the Royal Irtsh Academy.

V.-ONw TH EQUATIONw OF THE SQUAWB OP TEE IDFEUNCEs5 OF A

BiruAnD c. By Jont Cagr, LL. D, M. B. I. A., Professor of Mathematics in the Catholic University of Irelad.

[Read April 13dh, 1874.]

TEa following method of finding the equation whose roots are the squares of the differences of the roots of a biquadratic given by its general equation, with binomial coefficients, has been in my posseion for some years. It occurred to me, while reading Professor Roberts' solution of the same question, published in Tortolini's " Anali di

Matematica." As it is, I believe, shorter and more elementary than the solutions hithero published, it may be deserving of the attention of mathematicians.

I. Notation.

Let (a, 3C, c, 4 , f z, 1)4 = 0 be the quartic, then we shall denote b6 - ac, the discriminant of

(a, bc,, x, 1,) by H; ad + 2V3 3abc by 0;

o is evidently= , when A is the discriminant of the cubic

(a, 6, c, d, Xx, 1)3, and the vanishing of G is the condition that-the roots of this cubic may be in arithmetical progression; we shall also denote the quadratic invariant of the quartic, as- 43d + 3eS by A, and its cubic invariant or catalecticant

ace + 2bcd4 _d' - eb6 - c by Is;

then, since G1, H1 I, I, are functions of the differences of the roots, we have at once, by taking a = 1 and 3 =06, the well-known theorem

G2 =4H3- I3- rf.H(1

II. Eulkr's Reducii Cubic.

Let the quartic (a, b, c, , e, (x, 1)4 =0 be deprived of its second term,

and it beeomes, making a- 1,

xt - 6 + 4 Gz It - 3P = , (2)

and Euler's reducing cubic is

y93Hv'+3(H' - ) = ?- (3)

This becomes by changing y into y + H, that is, by taking away the second term, and making- use of (I)

I, 13

4~~~~~~~~~4

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CABBY,- O the Biquadratic. 41

i1. Equaion of Diferanc.

If xa,, Wx, X3, x4, be the roots of (2), and v12, v,', v3 the roots of (3), wea have by Ruler's solution,

( - x2) - 4(vz2 + v32) + 8v,v3.

Hence, if s be a root of the required equation, and a,, a,, a. the roots of (4)

= 4 (al + a3 + 2f) + 8 V'(,TH)(a,+ )

.s - 4 (at + a BH) =64(aaa,+ (a, + a,l+Ri.

Sow, by equation (4)

as + a, = -a, and aza, = _-.

Hence, making these substitutions, and putting y for a,, we get

16r, ( + 4y)' - 16ffz +--- =O. (5)

The question is now reduced to the elimination of y between (4) and (5), which io easily performed, as follows. From (4) we have

- _I2+I3 =4!; (6)

and eliminating in succession

I3 and y2 from (5) and (6),

we get the two quadratics,

48' - sy - ( - 16Hs + 16I,)=O (7)

8sy2+ (s'-16fxs+41) +121I-O. (8)

Again, eliminating y from (7) and (8) we get

_ -16fe +l161x + 7213 (9) Y

4 -w 96.M + 241s;

and substituting this yalue of y in (7) we get the required equationu-.

t..4811+8(IV2+ 96H')t - 82 (320'2 + 48.T,H+ 4513)s' -J8 (7,s' - 3841,11 + 28813H7) - 3 (6$1i+ 44xH+ 5123)s

+256 (I-271-3)= 0.

In the preparation of t Paper, ProfessorJall's MKemoir on the Solution of the Biquadratic, published in Volume in. of the "1 QuArterly Journal of Pure and Applied Mathematics," has been of much use to me.

5. I. A. PROC., s'a. ?I., VOL. It., SCIENCZ. G

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