746 Proceedings of the Royal Irish Academy. 



would be imaginary ; and therefore (ha\ing regard to the prefixed 

 minus sign) the expression Q- - QPP' would be negative, whatever 

 real value be given to x. But for that value of x which makes 

 P = 0, as also for that which makes P' = 0, this expression is posi- 

 tive ; hence it is impossible, in the case supposed — the case, namely, 

 in which A and C are both minus, that the sign of JB"- - 4 A C can ever 

 be minus. Whenever, therefore, C is minus, whichever be the sign 

 of A, the equation P = must have a pair of imaginary roots. 



3. If £-<4:AC, that is, if the roots Ti, r., of Q^-2,PP' = 0, are 

 imaginary, the sign of _5--4^C'will be minus; and therefore, the 

 sign of the product of the squares of the differences of the roots of 

 (2), or P = 0, must be plus, which can be the case only when all the 

 roots of P = are real. 



And in this way are established the theorems arrived at in a very 

 diiferent manner in my Paper " On the Imaginary Eoots of IN^umeri- 

 cal Equations."* The theorems themselves, as here arrived at, 

 are but so many inferences from the property which it was the main 

 purpose of this communication to prove, namely, as shown above, 

 that 



B'-AAC 



A^' 



or, which is the same thing, that (rj - ro)' is equal to 



- 3(Pi - R,)\R, - E,)\P, - P,y, 



where Pi, Po, P^, are the three roots of the cubic equation P= 0, 

 and ri, r^, are the two roots of the quadratic equation 



(2- - 3PP' = 0, or Ax' + £x+ C= 0, 



deduced fi'om this cubic. And although particular examples are 

 never necessary to verify a demonstrated general truth, yet as such 

 examples are often acceptable illustrations of theory, I shall here 

 subjoin one or two. 



1. The roots of ^3+ \0x^ + 31a; + 30 = are 



-2,-3,-5; 



and the expression Q; - ZPP' is 



Ao^ + Px + C = "ix- + 40.?; H- 61 ; 



.-. B" = 402 ^ 1600, and 4^6' = 4 x 7 x 61 - 1708, 

 — 1 08 

 and the difference, -S- - 4^1 6', is - 108, and — — - = - 36. 



o 



The differences of the roots of the equation are 

 - 2 + 3, - 2 + 5, and - 3 + 5 ; 



* ride Proceedings R. I. Acad., vol. x., Series I., p. 343 (1866-69). 



