118

ON CUBIC EQUATIONS

TRACT 28.

honour of sending to him, in a letter of the 26th of thesame month ; and that learned gentleman has since commu-nicated to the Royal .Society his said formula, together withhis own investigation of it, done in his usual verv accuratemanner. S.nce that time I have seen, in the Memoires deVAcad. for the year 1743, four expressions similar to theabove, given by JMr. Nicole, for the purpose of summingcei-ain terms of a binomial raised to any power, but unac-companied with any appearance of the idea of thus reduc-ing the one case of the cubic equation, to the other.

129. It is hardly necessary to remark, that any general

series, of each of the above four forms, is summed bymeans of the sum or difference of the roots ol these twoequations, x 3 — ± c 7 )x = 2b ; and that, by substi-

tuting particular numbers for b and c, we may thus sum asmany series of those forms as we please.

130. Ex. 1. We may now illustrate these formulas bysome, examples. And first, in the equation v 3 — 15x = 4.Here Qb — 4, and 3 l/[b % + c') = 15, consequently b = 2,and c z — 5 3 — b z — 125 — 4 =r 121 = 11% and

X = J/(c + b) — */{c — b) - v"l 3 — $s9 = '2712508, the rootof the equation x 3 — 3 \/'.b — c 3 )x = 2b, orx 3 + 3J/1 17 x = 4. And, as b is less than c, this equationbelongs to the two series in the latter case for finding theleast root. Hence, the terms of the two series agreeingwith the positive and negative terms of the series in Art.106, they will stand thus:

By the 1st series

By the 2d series

a =* *3333333c as M, 0 O 0 130

()t>3

4 h ^ 512

v? ~ ^ m

series — —x = +r

log. 1-5229217- 0-2081282

- "f-7317499

b = -002040(5D = -0000007

■002041346 512

log. 3-3099068- - 0-2088282

- Ml 87350

the same root.

l/c 3 ^121

series =-f *0033016-x =- -2712508r = — *2679492

5.192000

-'■" 19-08•'ao',9452 the least root.

Agreeing with the same root found in Ex. 4, Art. 106.