40 Proceedings of the Royal Irish Academy. 



Y. — OjS" the Equation of the Sqfaees of the DiEFEEEifCEs of a 

 \J BiQiJADKATic. By John Casey, LL. D., M. R. I. A., Professor of 



Mathematics in the Catholic University of Ireland. 



[Eead April 13th, 1874.] 



The 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 possession 

 for some years. It occurred to me, while reading Professor Roberts' 

 solution of the same question, published in Tortolini's " Annali di 

 Matematica." As it is, I believe, shorter and more elementary than 

 the solutions hitherto published, it may be deserving of the attention 

 of mathematicians. 



I. Notation. 



Let {a, I, c, d, e, '^ x, 1)* = be the quartic, then we shall denote 

 h"^ - ac, the discriminant of 



{a, I, c, \x, l,y-by H ; 

 a-d + 2b''-3ahc by G ; 



G-' is evidently = -^ — ■, when A is the discriminant of the cubic 



(a, h, c, d, Tx, ly, 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, ae-4ibd + 3c- by Z,, and 

 its cubic invariant or catalecticant 



ace + 2bcd- ad' - e¥ - (? by I^ ; 

 then, since G, S, L I3 are functions of the differences of the roots, we 

 have at once, by taking a = 1 and 3 = 0, the well-known theorem 



II. Elder'' s Reducing Ciilic. 



Let the quartic {a, I, c, d. e, "^x, 1)* = be deprived of its second term, 

 and it becomes, making « = 1, 



x'-6Kx' + iGx^L-3B:'=0, (2) 



and Euler's reducing cubic is 



f-2.m/^^[^ir-' - il ) ^ - T = °- ^^^ 



This becomes by changing y into y + S, that is, by taking away the 

 second term, and making use of (1) 



L h 



