"44 Proceedings of the Royal Irish Academy. 



LX. — On a Certain Helation between the Qttadeatic Expression 

 (^ — 3FF', AND the Peoduct oe the Sqtjahes of the Differences 

 OF THE Roots of a Cubic Euitation. By J. E,. Young, formerly 

 Professor of Mathematics in Belfast College. 



[Eead June 26, 1876.] 



Let ai^ + 2}.v + q = (1) 



be the equation which results from depmning the cubic equation, 



a^ + Aox"- + AiX + Aq = (2) 



of its second term ; that is, let (1) be the equation which arises from 



diminishing each of the roots of (2) by -- ^2 ; then, conformably to 



o 



the notation of my former Papers, 



I* = a^ + fx + q^ 



Q = 3r^+^ .-. Q- = 9a;* + Q2^X' + p^ 



P' = 3x 3PF' = 9x^ + d^^xr + 9qx 



Or - 3PF = - Zpx" - 9qx ^f 



JS'ow, the square of the middle co-efficient of this quadratic ex- 

 pression, diminished by four times the product of the extreme 

 co-efficients, furnishes the 



Ptemainder, 81^- + 12^ = 3 (27^- + Af) ; 



which (with changed sign) is three times the product of the squares of 

 the differences of the roots of the equation (1). [Theory and, Sohdion 

 of Equations, p. 410). [This is proved independently at the end]. 



But the differences of the roots of the equation (2) are the same 

 as the diffei'ences of the roots of the equation (1) ; because the roots 

 of (2) are no other than the roots of (1), each increased by the same- 

 quantity ; namely, by the quantity - - A^. 



o 



Calling this quantity a, the equation (2) is 

 {x + a)^ -V p {x + a) + q -0 ; 

 and the expression Q- - 3PP', for this equation, is 

 (2- - 3PP' = -3p {x + a)- -9q{x + a) ^p" 



= - 3ps^ - ifipa + 9q)x - Zpa^ - 9qa +j^*. 



