into Conjugate Factors. 281 



which we see reduces to the preceding expression for the roots 

 of a cubic when Xc^^-O. 



If we were to take an equation of the fifth degree, and to 

 proceed as above, we should obtain three expressions of the 

 third degree, as the expression (B) of the second degree is 

 obtained : atid if from the double of one of these the sum of 

 the other two be taken, and an obvious division performed, 



we shall get an expression for ;-, the product of the re- 



12 3 



maining two roots x^^ .r^: and, determining F as before, the 

 roots themselves are found to be 



2 



•T XyX(^-\-X]XQ-]rXc^Q Q ( > 

 of which the irrational part may be written 



-f- ^1 [Xc^ + Xq) + Xc^Q ^ \ * 



and thus the general form of the expressions for two roots of 

 any equation, when the others are found, is sufficiently indi- 

 cated. It is probable however that, beyond equations of the 

 fifth degree, these formulas would not be much more commo- 

 dious for actual numerical computation than those equivalent 

 ones in which minus the product of the two sought roots is 

 introduced under the radical, in the form in which it is imme- 

 diately obtained from dividing the final term of the equation 

 by the given roots with changed signs ; the formula in this 

 way being immediately suggested by the expression (1), from 

 which indeed what is here done might have been derived. 

 But my chief object has been to show how the conjugate factors 

 (A) may be turned to account in a particular inquiry ; as we 

 see that, from these, the form (1) has been itself obtained. The 

 reader will at once perceive how the term cofijugate factors 

 has been suggested ; and I would here venture an opinion that 

 the same term might with propriety be employed, instead of 

 congeneric, in certain equations related to one another in a 

 somewhat similar manner as these factors : and further, that 

 the expression conjugate roofs of equations seems to be unne- 

 cessarily restricted : all the roots of equations of an even de- 



