and Infinite Series. 44r 
found the roots of thofe equations in which that rule 
fails when it is applied fingly, that is, in what is called 
the irreducible cafe, or that in which c z is negative. 
And thofe feries are found by introducing another cubic 
equation having the fame values of b and c z as the given 
equation, except that in the new equation the value of 
c'~ is pofitive, while in the given one it is negative. For 
when c z is politive, the new equation to which it be- 
longs has- only one real root, and that root is always 
found by cardan’s rule; but the contrary takes place 
when c is negative, the equation having then three real 
roots, although they are not always determinable by 
that rule, becaufe the radical quantities can feldom be. 
extracted, on account of the fquare root of the nega- 
tive quantity which is contained in them. 
125. Now the general expreffion for the root by 
cardan’s rule being s + d - b + V ±c z +fi' b-V or 
>y \/ ± c z + b — v / v / i: c * 1 - b, if the cubic roots of each of 
thefe Ire extradled by the binomial theorem, as at Art. 
68, we lhail obtain thefe 4 forms ; 
1 . fi b + s/ + c 1 + b~s/ + c z ~ 2 fib X' 1 r — * — “ &c». 
2. s/b + V'- c* + %/b c- = 2 fib x : 1 + - 8cc. 
3. v 7 s/+ c 2 3 * +b — + c z -b = 4 x 
V 1 ' 3 
3.6*’ 
3.6.9 
4. ■fifi—c 1, + b — \/y/~ c x — b — 
2 1 ? 
I 2 .C fc 
vs x : " 7 + ~ 
4f.~ 
+ Sec.. 
&c. 
3 • & • 9 fl 
126. Of. 
