1815.) Scientific Intelligence. 31h 
negative quantity whenever the equation can be reduced by Cardan’s 
rule. Hence, that the roots may be both positive or both negative, 
, ey : 
we must take the sign of ( — x) contrary to the sign of 
; ee ’ b 
( fi — =) ; so that the equation in this case will be = (& — | 
ha ie fick cag A) ay ({-+): ; sonslict 
aN ( 7 mee 4 Acmaidiia vA Gay)? bata the equation be 
: : - BAS A 
long to the irreducible case, the quantity (¢* — =) being then po- 
sitive in itself, both parts of the roots on the left hand must have 
; ‘ é 
the same signs. Hence the equation will be + (# — 3) x El a/ 
ies 2 a3 ) : : 
See eee = et EW 
; ;)=+ / ( 7 = and by proceeding with these two 
equations as Mr. L. has done in the letter alluded to, we obtain 
t za b 
from the first of them, — out 7TCR3e 
Ss 
SYec 4 oe 3 = 1 for 
ls Sy ems Y, the same as the root given in my former 
J 2 
letter; and from the second we get — : +4 / S _ 
3Sfc ce 03 A 
wer J+ — > the same with Mr. L.’s root. 
With respect to the two roots connected with w and v, I have 
only to observe, that they are obtained in the very same manner as 
that connected with ¢, ouly there is no ambiguity in the two quan- 
col > 
its b b : 
tities (x? — =) and (v? — xe the former being always a pasitive, 
and the latter a negative, quantity. 
Before I take leave of this subject, it may not be amiss to observe, 
that by inspecting the formule for the cube roots of the two ima- 
' Pe ry ame ee IN crn 
ginary quantities VA; + / pach V2 a /+ ahirns 
when the given equation x? — l x = c is irreducible, it is manifest 
that their sum will always be a real quantity; for the imaginary 
parts in the roots of the first of these quantities are the very same as 
the imaginary parts of the roots belonging to the second, but having 
contrary signs. It likewise appears that the real quantities arising 
from taking their sum will always be the ¢hree roots of the given 
cubic equation, 
This appears to me to be a more direct and satisfactory demon- 
stration, that Cardan’s theorem, though apparently an imaginary 
uantity, exhibits truly the rogts of equations belonging to the irre- 
ucible case, than the one generally had recourse to, viz. to expand 
each part of the root in an infinite series by means of the binomial 
theorem, 
It likewise appears from these formule that whenever any one of 
the roots of a cubic equation admits of a finite value, the two parts 
of Cardan’s theorem are both perfect cubes, 
