750 Proceedings of the Royal Irish Academy. 



and the square of the difference of the two roots Vi, Ti, of the quadratic- 

 equation 



Qi - 2>PP' - ; that is, oi Ax" + Bx ^ C = , (2) 



the relation — namely, 



A 7P—4AC 



=1 (r,-r,y = —JT- =- 3 (-^1 -^^y {B.-H.y {B,-B,y, .... (3) 



can subsist only when the equation Q' - 3FF' = has two roots ;. 

 that is to say, only when ^ is a significant number. If A be zero, the 

 equation (2), being then of only the first degree, has but one root, 

 and the first member of (3) is nugatory ; but the second member re- 

 mains significant ; it is £' ^ As'. But if C be zero, and A a signi- 

 ficant number, one root (rj) of the quadratic equation will be zero :. 

 and the first and second members of (3) will then be 



A' . ,B- 

 — r,', and -^^, 



implying that when the quadratic equation Q' - 3FF' = is 



Ax^ + Fx=0, 

 tliree times the product (with changed sign) of the squares of the difPer- 



752 



ences of the roots of (1) is equal to —j, or simply to £^, or A-ri^, if 



the co-efiicient A^, in (1), is unity. 



For example : suppose we have the equation 



F = x"'+Sx"-6x + 4 = 0, 



where the second triad of co-efficients furnishes the condition 



Ai' - ZAiiAo = 0. 



The equation Q- - ZFP' = 0,[here, is found to be 



Ax"" + Fx+ C= "Tia? - 54«= ; 



in which the values of x are a; = 0, and x = ry^1. 



IN'ow, if each of the roots of the equation P- be diminished by 

 - 1, the second term will disappear in the transformation, which will 

 be the equation 



a.'3-9a: + 12 = 0; 



and the product of the squares of the differences of the roots of 

 this equation, when the sign of that product is changed, is (by p. 410, 

 Theory of Eg^uations) 4^^ + 21^, where, in the present case, ^ = -9^ 



