too

ON CUBIC EQUATIONS.

TRACT 2S-

sides of the last equation, we find thatI is = l +

2

2 . S

+

3.5.8

+

3 . 5.8.11

&C.

3.6 ' 3.6.9 * 3.6.9.12 ' 3.6.9.12.15Or, further, multiplying by 3, and subtracting 1, we have„ 2 3.5,3.5.8 . 2.5. 8 .11 .

~~ 6 "*“6.9^6.9- 12^*6.9.12. 15 C '

74. Also from 2 x : 1 — —-

, i> . o

last article, we find \l/2 —

2.5.8

3.6 ,9 . 12

- &c = 1/2 in the

, 2 2.5.8

3T6 — 3.6.9.12

&C = - + -

2.5

+

2.5.8.11

&C.

3 * 3.6.9 ‘ 3.6.9. 12.1575. In this case also, namely, c — b, the equation d —

l/{b — c)=%/b x : 1 —

2c*

35

3 . 65 *

&c, becomes 0 =$/b x : 1 •

&c.

3 3.6 3.6.9

And hence, dividing by %/b, and adding, we have

1 -T+3.6

2 , 2.5 , 2.5.8 „„

+ £ ' * o + ~c 'o 13 CfcC ’

3.6.9 1 3 . 6.9 . 12

the same as in the last article but one,

76. And by taking other values of b and c, or other rela-tions between them, any number of infinite series may beassigned, whose sums will be given by the two equations

fy(b±c) = */b x :l± ~

+ __ 2 __ 5 , . 3 _3 . 65 * ~ 3.6 . 951

And if b

be very great in respect of c, the two first terms of theseries wjll give the cube root true to many places offigures.

77. Hitherto is concerning one of the limits or extremecases only, namely, when c 1 = b z , or when the equation isx 3 = q = 2 b. And it has been observed, that the first gene-ral series for the three roots converges, in all the cases ofthe equation x 3 — px = q, or x 3 — 3ax — 2b, in which a 3 isnot greater than 2b~. But a 3 may be any real quantity, notgreater than 2 b z , and so it may be either less than, equal to,or greater than b 1 .

78. When, in this equation, a 3 is less than b l , then c 1 ispositive, and less than b z , and the first series gives the onlyreal root, without any change in the signs of the terms.