and Infinite Series. 39 3 
10. Fir ft, we hence find, that when the equation has 
three real roots, the fign of the fecond term is always - ; 
for the coefficient of that term, or p is = - | r 2 — e z , 
which is always negative when r and e are real quanti- 
ties. And confequently when p is politive, the equation 
has two imaginary roots, fince —p includes all the cafes 
of three real roots. But it does not therefore follow, 
that when p is negative, the three roots are always real; 
and indeed there are imaginary roots not only whenever 
p is pofitive, but fometimes alfo when p is negative: for 
fince p is = - |r 2 - e 1 in all the cafes of three real roots, it 
will be p — — |r 2 + e 1 for all the cafes of two imaginary 
roots; and it is evident, that p will be either pofitive or 
negative, according as e 2 is greater or lefs than |r 2 . 
1 1 . But to find the cafes of — p when the roots are all 
real, and when not, will require fome farther confidera- 
tion; and in order to that it muft be obferved, that e z 
ought to be pofitive and lefs than |r 2 ; but the limit be- 
tween the cafes of real and imaginary roots is when <f =0, 
or e - o ; and then p becomes = - |r 2 , and q = \r l ; con- 
fequently then fir 6 , which is 
= fir 6 , that is, when e is =0, then Jpi' is = fip , and con- 
quently when is lefs than , the equation has two 
imaginary roots, otherwife none, the fign of p being -» 
Thus 
