and Infinite Series. 
417 
71. That r be lefs than b z , or the foregoing feries be 
proper to be ufed, a or \p muft be a negative quantity ; 
for if it be pofitive, then d = b z + a 7, will be greater'than 
b x . But for this purpofe a cannot be any negative quan- 
tity taken at pleafure ; for if it be fo taken as that a 3 be 
greater than 2 b\ then fhall — c 1 - a> — b z be greater 
than b z . And hence thefe feries converge only in fome 
of the cafes of three real roots, and in fome of thofe that 
have only one real root, namely, from the 16th form to 
fomewhere between the 1 2th and 1 3th forms in the ge- 
neral table Art. 30. when b as pofitive, and confequently 
it includes fome cafes both with and without imaginary 
roots. But that in all the cafes, the firft feries 
s + d — i\/b x : r Sec. is the greateft root, as 
3 . O 0 0 7 
will Hill more fully appear by confulting Art. 83.. 
72. Now, in the firft place, when a - o, or c = b, 
which is the limit, or 16th cafe in the table. Art. 30, 
the equation being ar 3 = q - 2 b, then the only real root 
is s = d/b + c - V ib — \Iq - f/b x : i + \ \ + See .. 
3 3.0 
Hence alfo, dividing by J/b, we have 
•y 2 = 1 + - — + 
3 3-6 
See. 
3.6.9 3.6.9. 12 
But in this cafe alfo the root is 
— - - 2,5,8 &c. And 
73 
3 + d = 2 fi/b x : 1 - 
3.6 3.6.9.1a 
confequently 
