98

ON CUBIC EQUATIONS

TRACT 28.

Of the Roots by Infinite Senes.

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 + d and — ± ~s~v / — 3, or

t/(b + c) + %/'b — c) and

c) + Yfi - c)] ± if - 3 X [V(6 + <■) -l/{b- <•)] •For, by expanding i/[b ± c) in an infinite series, we shallevidently have all the roots expressed in such series

68. Now s == %/(b +c) =^x:l + 2 i,

2 .

3 . 66 2

and d=l/{b-c)~,i/bx: 1 ~

2t 2 , 2 . 5r3 q

— 4- ■ iSw C >

54“ 3 . 6 . ir't'3 ’

Hence s + d — 2 l/b x : 1 •

2 c 2

3.6. 9432 . 5 . 8t4

Sec,

&c,

3 ■ 64 2 3 . 6 . 9 . 1244

for the first root, as it was found by Mr. Nicole, in theMemoires de VAcad. 1738. Also

&c. Therefore

,2 c

~ d = ^x--i +

s + d

~2~

2 ,5 V , 2 . 5 . 8 . 11 rA

3.6. 94* 3.6.9.12 . 1544

) C ,__ 2 • 5 - 8c< & c

y _ J X . 1 3 . 61,1 3.6.9. ,0, '“

‘S l

C *J-

, 9 • 5c * .

■ X : T+'« £ nw +

1264

2 . 5 . 8 . 11 c 4

i &C,

3/6 2 ” ' 3 1 3.6.94» 1 3 . 6 . 9 . 12 . 1544

for the other two roots, which were given by Clairaut, inhis Elemens d’Algebra.

69. Hence again it appears, that when e 1 is positive, thesetwo latter roots are imaginary; for then the factor ■is imaginary. And that those roots arc real when this c z is

negative; for then this factor becomes ——

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 c z ; for the series contain the letters as adaptedto the positive sign only.

70. These series are proper for those cases only in whichc 1 is not greater than b 1 ; for if c 1 were greater than 6% they