if ,1

■ %■

120

ON CUBIC EQUATIONS

TRACT 28.

And these values of the greatest root are nearly the samewith that found in Art. 110.

134. But, in Art. 61, the same root was found to beu- 1 — ; hence we obtain the sums of these first two

particular series; and by the addition and subtraction ofthese two, arise the other two following them, namely,

l + v'6 + V(5 + V'2) + V(5-V2) , 2.5.8 . 2* .

4J*/5 1 3.6.9.12.5“ <XC ’

3 + a /6-V(S+V'«)-V(5-a/2) 2.2 2.5.8.11.14.23

4*/5 3 . 6 . 5‘ 2 *

a + V 6

2 %/5

= 1 +

2.2

2 . 5 . 8 . 2 *

3 . 6.9.12.15.18.5“2.5 . 8 . 11 . 14.23

3.6 . 5 2

3.6.9.12.5“2.2

+V 2 )+v(s_rVl> - i _

23/5 1 3 . 6 . 5*

3.6.9. 12 . 15 . 18,5'2.5 . 8.2 a

-&C;

-&C;

— &c.

3 . 6 . 9 . 12 . 5“

And the last but one of these equations agrees with onefound in Art. 112.

135. Ex. 3. Also, in the equation x 3 — 12.r = 9, We haveQb = 9, and %/(b z -f- c z ) = 4; consequently b == f, and c 1 —4 3 —b !i = 64—! y = i^. s , which being greater than b z or y,this case belongs to the second class of series, or that of the

Jeast roots. Now here x =£/(<-' + b) — %/{c — b) = ^ + 9 —

= 3 /ll-U4378 - 3/2-114378 = 2-2316619 -

1*2834950 = -9481669 = the root of the equationx 3 — $%/{b z — c 1 ) . x = 2 b, or x 3 + 3’/¥ . x — 9. And theterms of the two series being found as in Art. 113, namely,A+c + e + &c = -34051, and b + d + r -|- ike = -03071,.4 b , . 36

als0 .7^ beln g

3/25QBy the 1st series

, we shall have

0-7082798

•34051 - log. 1-532129936

i/350 ' _

series =— 1-739441 - - 0-2404097x =+ 0-948167

— -791274 the least root.

By the latter series

0-7082798

•03071 - leg. 2-487279S36

' _

series =+ -1568771 - - 1~1955596

x =_ -9481669

— -7912898 the same root.

Which nearly agree with the same root found in Art. 113,