30 Proceedings of the Royal Irish Academy. 



is obvious that the foregoing condition can never be satisfied unless the 

 coefficient A^ is negative.] 



Whether, in the reducible cases of the equation a;^ + yl^^ + ^o = 0, 

 that is, in the cases in which 21 Aq + 4:Ai is not less than zero, the 

 formula [5] may, or may not, advantageously replace the old formula 

 of Cardan, must be left for the discrimination of algebraists to deter- 

 mine. Few persons, however, are likely to prefer a formula to the 

 method of continuous approximation, whenever the numerical values of 

 the roots of an equation of higher degree than the second degree are 

 the only objects of search. 



(4.) Whenever it happens, in this ease of an incomplete cubic equa- 

 tion, that (9^o)^ + 12^l^ 01') which is the same thing, that 27^o* 

 + 4J.1' is zero, the two values of r, as expressed in the foregoing for- 

 mula, will be equal values ; and, as already proved in the Paper before 

 referred to, these are also two roots of the cubic P = ; hence, when- 

 ever this condition has place, the incomplete cubic equation has a pair 

 of equal roots, each of these being, 



'^~ 6^1 ~ 2Ai' 



It may be well to observe here that the case of equal roots in any 

 cubic equation, is excluded from the general investigation at article (2J. 

 The transformed equation, in x' , of which the first three terms are 



A^x'^ + (3^3r + A^)x"- + (3^3?'2 + 2A.^ + Ai)x', 



is there considered to have a significant coefficient for its third term ; 

 since if the expression 



ZA^r'^^2A^r + A, 



were zero, the first member of this transformed equation could not be 

 made a complete cube by merely modifying the coefficient of x'^, as, 

 in the investigation alluded to, is assumed to be practicable. 



When the indicating quadratic (Q-- 3 PP' = 0), as we may call it, 

 has two equal roots, and, therefore, the cubic P=0 has the same two 

 equal roots, the above- written coefficient must be zero ; in which case 

 the transformed equation becomes simply 



A^x'^+{ZAzr^A^)x"' = 0, 



of which the three roots are 



, , , ZAzr + Ao 

 X =K), X = Q, X • 



