Mr. Graves on Cubic Equations. 217 



their metallic surfaces) proceeds as a natural consequence 

 without the aid of any hypotheses. The various action of the 

 insulated pile is still enigmatic, but might possibly depend on 

 this, that with a bad conducting fluid the amount of conduc- 

 tion of the pile becomes greater, since then the electricity 

 probably penetrates even between fluid and metallic surface, 

 which might be less the case if the conducting power of the 

 fluid approached more to that of the metal ; a subject more- 

 over on which special experiments are still desirable. 

 [To be continued.] 



I 



XXVIII. A New and General Solution of' Cubic Equations. 



By John T. Graves, of the Inner Temple^ Esq., M.A.* 

 N the ordinary books of algebra, (so far, at least, as my 



limited acquaintance with them extends,) where cubic equa- 

 tions are discussed, the cases o^ real coefiicients only are con- 

 sidered, and different methods of solution are given in order 

 to effect the separation between the constituentsf of the roots 

 in different cases. I have obtained a symmetrical solution 

 of the equation 



^H(x+ v'^rA)a; + i«'+'v/^v = (1.) 



which presents the constituents of x in an explicit form in all 

 cases. This is all that is wanted, for the solution of a cubic 

 equation of the general form. 



(«i+ >/=i /3i)y + («2+ v^ /32)y+ («3+ v=i ^s)2/ 



+ a4+'/-l/S4 = (2.) 



may easily be made to depend on the solution of a cubic 

 equation of the form (1.) ; and, from the nature of the rela- 

 tion between the transformed equation (1.) and the original 

 equation (2.), the constituents (x, A, jw., v) of the transformed 

 coefficients (x + a/— 1 A, jU-4- V— 1") are easily determinable, 

 supposing the constituents of the original coefficients to be 

 explicitly given : and if the constituents of x be determinable, 

 those of 7/ can easily be determined. 



The limits of this Magazine do not permit an exposition 

 of my process, which I intend hereafter to communicate at 

 length through some more appropriate medium. It consists 

 in an analysis of the following formula for x. 



where p — x+V— 1^ and q = /x 4-^—1 h and where the 



• Communicated by the Author. 



t I call ec and /3 the " constituents" of the expression et + a/—\ /3. 



