itefcilution of Algebraical Equations. '89 
the unknown quantity b may be found, which being 
fubftituted for its value (b) in the preceding equations, 
from the equations thence enfuing may be found the 
unknown quantities a and c, and confequently the re- 
folution of the given biquadratic x 4 + qx~ -rx + s=o. 
From the fame principles can be deduced different re- 
folutions of the above-mentioned biquadratic x^+qx^ 
rx+s-o. 
3. I. Let x~aP/p + bP/p\ then will the equation 
free from radicals be x' 1 " — ib H px K — dh. inb n ~'a' ! px’ t ~' 1 — 
x inb n - 1 apx n - 1 - ’" - I ™-* x anb^afx^ 
1. 2.3.4 r 1.2.3.4.5.6 t 
n X tr — 1 x // — 4 
r - g X 2 nb n -* a % px n -* ....... 
1.2.3.4.5.6.7.S 
X ^ X 73 - 9 . ?Z 76 .. x 2 na ™-i bpx = a «p_ b y. 
I, 2. 3. 4.5, 6 . 7. ..-. 2 ^ — 
This ‘equation may be deduced from the following 
principles. Let oc, ft, 7, S, s, &c. be the 2 n roots of 
the equation z tn — 1=0, then (by Prop. xxm. of my 
Meditat. Algebraicae) the equation free from radicals will 
‘be the product of the following quantities {x—axP/p — 
boCVp) (x -afivp- bftP/p) (x -a y P/p- b/p/ff) (x - 
aSP/p — b/P/ s p r ) ( x — azP/ i p — bpP/p 1 ) 8cc. = o : multiply 
thefe quantities into each other, and from the refulting 
product, by Prob. HI. of the Meditat. Algebr. eafily can 
Yol. LX 1 X. N "be 
