39 s jDr. hutton on Cubic Equations 
is to be connected with —\r by the figns + and - to 
compofe the other 'two roots, will be a real quantity 
when thofe roots are real, but an imaginary one when 
they are imaginary, lince the other part (-{r) of thofe 
two roots is real by the hypothecs. ,Let this fupplemen- 
tal quantity be reprefented by e when it is real, or eV - 1 
or \/ -e~ when it is imaginary : we fhall ufe the quantity 
e in what follows for the real roots; and it is evident, 
that by changing e for e\/ - I, or e" for —e L y that is, by 
barely changing the fign of e 1 wherever it is found, the 
expreffions will become adapted to the imaginary roots. 
Hence then the three roots are reprefented by r, and 
- \r + e, and —\r~e \ and confequently the three equa- 
tions, from whofe continual multiplication by one ano- 
ther the cubic equation is to be generated, will be 
x - r — o, and x + \r - e — o, and x + \r + e — o. 
9. Let now thefe three equations be multiplied 
together, and there will be produced this general cu- 
bic equation wanting the fecond term, namely, 
x 3 !&■* x - r . - e z = o, or at 3 z]C x = r . \r z — e z , hav- 
ing three real roots; and if the fign of <r be changed 
from - to +, it will then reprefent all the cafes which 
have only one real and two imaginary roots : and from 
the bare infpeftion of this equation the following pro- 
perties are eafily drawn. 
10. Firft, 
