9&

ON CUBIC EQUATIONS

TRACT 2f.

Of the Roots by Infinite Series .

67. Another way of assigning the roots of a cubic equa-tion, may be by infinite series, derived from the fore-going formulae, namely, s -j- d and — s -~ ± S -~”f — 3, orl/(b -f c) -f l/'b — c) and

-t X' il/{b : +e) +%/fi-cy ±|v/-3 x\?/{b-\-c)-l/{b-c)].For, by expanding ± c ) in an infinite series, we shall

evidently have all the roots expressed in such series.

'' , i , , , , , c 2c 2 ,2 . 5c3

V'fi + c) =yb X : 1 +

68. Now s =

and d = i/[b — c) — yb x : 1 ■

3 . 6i» ■ 3 . 6 . St’Sc 2(- 2 2 ■

~36 3 . 66* 3.6-963

2c 2 2.5. 8t4

&C,

&c,

&C,

Hence 5 + d = X : 1 - 37^-37 6 . 9 . 12A4for the first root, as it was found by Mr. Nicole, in theMemoires de VAcad. 1738. Also

7 9c

■d = —- X

yi> a

T

2.5c 23 . 6.9p

+

2.5.8. Ilc43 . 6 . 9 . 12 . 1564

Sec. Therefore

, , , 2c 2 2.5.8 C 4

^ X : 1 3 . 6b 2 3.6 . 9. 1264 C ’

+ ^x : , + -l^l + ^_l- 8 :.^-&c,

~ yb* 7 3.6.9t 2 3 . 6 . 9 . 12.1564

for the other two roots, which were given by Clairaut , in

his Elemens d’Algebre.

69. Hence again it appears, that when c z is positive, these

two latter roots are imaginary; for then the factor -is imaginary. And that those roots are real when this c* isnegative; for then this factor becomes c -~>

a real quantity. But in this last case, the sign of everysecond term in the two series must be changed, namely, thesigns of the terms containing the odd powers of the nega-tive quantity r ; for the series contain the letters as adaptedto the positive sign only.

70. These series are proper for those cases only in whichc L is not greater than b z ; for if c z were greater than b z j they