TRACT 23.

AND INFINITE SERIES.

would all diverge, and be of no use : and the series proper

for the other cases, namely, in which c 2 is greater than b\we shall give below.

71. That £® be less than 6 2 , or the foregoing series beproper to be used, a or j-p must be a negative quantity ;for if it be positive, then c 2 — b 1 + will be greater thanb z . But for this purpose a cannot be any negative quantitytaken at pleasure ; for if it be so taken, as that a 3 be greaterthan 2b z , then shall — c 2 = a 3 — b 1 be greater than b 1 . Andhence these series converge only in some of the cases ofthree real roots, and in some of those that have only onereal root, namely, from the 16th form, to somewhere be-tween the 12 th and l3th forms, in the general table Art. 30,when b is positive, and consequently it includes some casesboth with and without imaginary roots. But that in all the

cases, the first series s -(- d — 2 fyb x : 1 — ——- &c, is the

greatest root, as will still more fully appear by consultingArt. S3.

72. Now, in the first place, when a~ 0 , or c ~ b, whichis the limit, or 16th case in the table Art. 30, the equationbeing x 3 = q — 2b, then the only real root is s =l/{b -|- c) =

consequently this is equal to the former series, or

Hence, by subtracting 1 — - — -——& c f r0 m both

which multiplied by 2 lyb, will also give the root of the sameequation. And hence, adding — + &c to both

\/2b = */q x :1 + -^ — y- 7 . T &c. Hence also, dividing

2.5.8

hyfrb, we have 1/2

3.6.9

3.6 . 9.12

73. But in this case also the root is

2.5.8

J + d = 2 l/b x : 1

Sec. And

3 . e . 9 .12

2.5.8

&c — 1 +

2 x : 1

&g = 1 / 2 ,

3.6.9.12

3.6.9

sides, we have, 2 2.5.8

3.6.9

2 . 5 . 8 . 11

3.6 3.6.9

3.6 . 9 . 12

3 . 6 . 9 . 12 .15

&C,