90 A Jsew Method of Resolving certain Equations:^ 

 Fpoki this we have the 



Rule. 



To find a>, when x^-{-4ax^-{-Gbx^-\-4cx+d—0, 

 (C) Find z, in the cubic, z^ — 6bz^-\-{iac-{-9b^—d)z 



- 12abc-{-2aU-{-2c^ =0. 



(D) Find Kjin the quadratic, aas^+ro; -{-0=^/0 ^ — -.x^ 



At 



2 



It may be seen that this solution might also be applied 

 to equations of the third degree ; for every such equation 

 may by simple multiplication by any quantity, {x-\-w) be 

 raised to one of the fourth degree, containing a variable in 

 its co-efficients. In the cubic (C,) if we destroy the se- 

 cond term by substitution, the remaining co-efficient in the 

 resulting equation is (4ac — 36^ — <Z);and,as (w) enters but 

 once into each of the quantities denoted by a, b, c, d, It is 

 evident that the co-efficient named, can never rise higher 

 than to the second degree. It may, then, be supposed to 

 vanish; and the given equation is reduced to a cubic of the 

 form, s'^^-f c'=o. 



Neither of these solutions presents any advantages in 

 practice. Whatever value they possess, must be ascribed 

 to the fact that they are apparently tiezo in their form, and 

 in the mode of investigation. 



New-Haven, 1822. 



