O19 
binomial form, or to the extraction ofa 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, 
2 —102a°y*?+5ary'=a, 
5aty—l0x?y?+y°=), 
into which the imaginary equation 
(x +V¥ —ly)®=a+V —1b 
resolves itself, may be transformed into two others which are 
of the forms 
: 57r—10r?+7° 
p°=r, and “110-538 
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 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, 
2° — 102° y’?+5ry*+r=4, 
Saty—102* yy? +y°+y=), 
it ought now to be the object of those who interest them- 
selyes in the improvement of this part of algebra, to inquire, 
H 
