Young — Product of the Squares of the Differences, etc. 751 



and 2- = 12. Hence, three times the product of the squares of the dif- 

 ferences of the roots of P = is 



(- 2916 + 3888) x 3 = 2916 = 54^ = B- = Ah'{'. 



If A and C be each of them zero, then B itself will he zero ; and 

 P will be a complete cube, or a complete cube multiplied by a 

 numerical factor. It was shown in my Paper (read November 9, 

 1868) that P being A^x^ + A-^x" + A^x + Ao, Q^ - 8PP' is 



(^2^ _ QA.A^) x" + {A,A., - 9A0A3) X + (^1^ - 3 ^0^2) ; 



and that the two conditions 



A^^ - 3A1A3 = 0, and A,' - ZA^A-^ = 0, 



necessitate the third condition 



A,A. - 9AoA, = 0, 



will be seen by transposing the minus term of each, and then multi- 

 plying the results together ; for we shall thus have 



AM2' = 9A,A,A2M ; 

 and, consequently, 



A1A2 = y-4o-43 : 



so that, when the two foregoing conditions have place, the expres- 

 sions Q^ and SPP' must be identical; and, therefore, P must be 

 of the form .^3 (^ + af. 



I shall now give a simple and direct proof of the property referred 

 to (already otherwise established) at the commencement of the pre- 

 sent Paper — namely, that if D^ represent the product of the squares of 

 the differences of the roots of the equation 



x^ + px + q = . . . . (1), 

 we shall always have 



Z>2 = -(272'+4y). 



Demonstration. — It is shown in the Theory of JEquations, p. 322, 

 that if Xi be either of the roots of the equation (1), all three of 

 the roots will be 



2'"""' 



-^x, + V ^- o -- -py-j.,-^^- 3--P 



Kow, the differences of these are 



^.-v(-3^-A'^'.w(-3^-A2v(-3^-^)....(2), 



4 ^ y 2 ' ■ ' V 4 



R. 1. A. PKOC, SER. 11,, TOL, II., SCIENCE. 3 Y 



