binomial form, or to the extraction of a fifth root of an ex- 

 pression in general imaginary ? And he conceives, that the 

 propriety of considering such extraction as an admitted in- 

 strument of calculation in elementary algebra, is ultimately 

 founded on this : that the two real equations, 



x b — 10 x 3 y 2 + 5 xy* — a, 



5x*y — 10*V + y 5 =&, 

 into which the imaginary equation 



(, r _|_ V —\ y) 5 — a _|_ V ~\Jj 



resolves itself, may be transformed into two others which are 



of the forms 



5r-l0r 3 + r 5 _ 

 p 5 =r,and ^ _ 10t , + 5t 4-«, 



so that each of these two new equations expresses one given 

 real number as a known rational function of one sought real 

 number. But, notwithstanding the interest which attaches 

 to these two particular forms of rational functions, and ge- 

 nerally to the analogous forms which present themselves in 

 separating the real and imaginary parts of a radical of the 

 n ih degree ; Sir William Hamilton does not conceive that 

 they both possess so eminent a prerogative of simplicity as to 

 entitle the inverses of them alone to be admitted among the 

 instruments of elementary algebra, to the exclusion of the 

 inverses of all other real and rational functions of single real 

 variables. And he thinks, that since Mr. Jerrard has suc- 

 ceeded in reducing the general equation of the fifth degree, 

 with five imaginary coefficients, to the trinomial form above 

 described, which resolves itself into the two real equations 

 following, 



x b ~ \0x 3 y 2 + 5xy 4 + x = a, 



5x i y—\Qx 2 y 3 + y 5 + y = b, 



it ought now to be the object of those who interest them- 

 selves in the improvement of this part of algebra, to inquire, 



H 



