into Conjugate Factors. 283 



Again : if a pair of conjugate roots of a biquadratic equa- 

 tion be given in the form «+ \//3, then it is plain, from the 

 same expressions, that we must have 



and consequently the values of the two remaining roots will be 

 2a-fj? ///2a+^\2 s \ 



which might indeed have been deduced from the form (1), 

 though I believe that form has not hitherto been employed for 

 this purpose. The same values, as furnished by the expres- 

 sions previously given, take the somewhat more convenient 

 forms 



I shall merely add, in conclusion, that, as far as equations 

 are concerned, the conjugate factors of <^{x) do little more than 

 express the fact that the roots of every equation of an even 

 degree, 



{x^-\-ax-\-b){x'^ + a-^x-\-b^ .... =0, 



may be exhibited under the forms 



F+i/{F2-6} 



&c. &c. ; 



a truth which, however obvious from the theory of common 

 quadratic equations, has never, I believe, been turned to any 

 account elsewhere. The conjugate factors here referred to 

 express obvious identities : they do not presuppose the solution 

 of a quadratic, but actually supply that solution, by aid of the 

 fundamental property that the coefficient of the second term is 

 the sum of the roots with changed signs, and the third term 

 the product. 



P.S. In my last paper (p. 194, line iS)for "quotient" read 

 " product." 



Belfast, March 8, 1849. 



