748 Proceedings of the Royal Irish Academy. 



Therefore tlu'ee times the product of the squares of the differences of 

 the roots of the equation P = is -^ 147. Also 



-147 , 



i\, ro heing the t^^o roots of 427j;-- 27ol.r + 4431 = 0. 



These two roots -woiild, of course, remain unaltered, although 

 the co-efficients of the equation Ajt' + Bx-\- C- were each multiplied 

 or divided by any number ; but the numerical result, W - 4^ C, would 

 be changed by such multiplication or division, and would no longer 

 express thi'ee times the product of the squares of the differences of 

 the roots of P == 0. The sign of the numerical result would, however, 

 be the same ; and therefore, for the purpose of ascertaining the 

 character of the roots of the cubic equation -P = 0, the co-efficients of 

 the quadratic equation Q;-2)PP'=0 may always be reduced to 

 smaller numbers whenever they have a common factor. Thus, in the 

 present example, we see that the co-efficients in the expression 

 Q- - 3FI" are each divisible by the number 7 ; so that we may 

 write 



61.C- - 393.-/: + 633 = A'x-^B'x + C, 



&W - 393A' + 683 = Air + B'x - C, 



and thus get 



B'^ - 4: AC = 15449 - 154452 = - 3 ; and ^ =(ri -^2)-; 



6P 



- 147 



this last result being the same as =(^1-^*3)^- 



It is obvious that if K denote the number by which each of the 

 co-efficients A, B, C, is divided, in any case, K- times B'- - 4:A'C' 

 will be ecjual to B^ - 4:A C ; that is (changing the sign), to three times 

 the product of the squares of the differences of the roots of the equa- 

 tion F = 0; in which equation, it is to be observed that the co-effi- 

 cient of x^ is tmity. If the co-efficient of x^, in the proposed cubic 

 equation, be A3, a number different from unity, then it is 



A, . ., W--4AC 



-— (ri - r.^', or , 



which, with changed sign, is equal to three times the product of the 

 squares of the differences of the roots of the cubic equation; that is, 

 of the equation 



P = A^x^ + Aox- + AiX + Ao = 0; 



because the co-efficients A, B, C, as deduced from this equation, are 

 each A^^ times what they "would be if the co-efficient of x^ were 



