and Inftnite Series. 2 95 
trary of this will happen; for then \r is lefs than <?, and 
confequently \r lei's than e\ and fo \r z - f a negative 
quantity, and therefore the product r . \r l - ~e\ or q, will 
have the fign contrary to that of r; that is, q and the lefs 
roots have different figns, and confequently q and the 
greateft root the fame fign, fince the fign of the greateft 
root is always contrary to that of the other two roots. 
14. Moreover, when <7 or r fr z — e’ is pofitive, then r 
denotes the greateft root; for then \r z is greater than <?*, 
or \r greater than e , and r greater than either —\r + e or 
~ * r ~ e ' when q or r . ^r z — e 1 is negative, then r re- 
prefents one of the other two roots in the equation; 
fince then * is greater than ±r, and -±r - * greater than 
r. Laftly, when q is between the pofitive and negative 
ftates, or q = o, then r ought to be neither the greateft 
nor one of the lefs roots, if I may fo fpeak, that is, two 
of the roots are equal, and the third root = o, fince then 
\r z muft be = e z , or \r — e. 
15. Hence it appears, that the fign of p determines 
the nature of the roots as to real and imaginary, and the 
fign of q determines the affedion of the roots as to pofi- 
tive and negative. Let us illuftrate thefe rules by a few 
examples. 
16. The equation - 9 x = IO has all its three roots 
real, becau fe/> = - 9 is negative, and |/1 3 = f = 27 i s 
Vo l. LXX. g g g greater 
