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 he1—4/6; hence we obtain the sums of these first twoparticular series; and by the addition and subtraction ofthese two, arise the other two following them, namely,

2.5.. 8 . 2* 0

&C ;

3 +a/ 6+ 3/(3 + V 2) + V(5-y2) _43/5 ~

H-a/6-1/(5 + aA)--1/(5-a/3)

41/5

2.2 2 . 5,8 . 2 2

23/5 T 3 . 6.5 2

3/(5 +a/2)+ 1X5-a/2)

3 . 6.9 . 12.5*

2.2 , 2.5.8 . II . 14 . 23

3 , 6.5 a 3 . 6.9.12.15.18.5«"2.5.8 . 11 . 14.23

3 . 6.9 . 12. 5 *

2.2

"3 . 6.9 . 12 . 15 . 18.5“

2.5.8.2-

-&CJ

&CJ

— & c.

23/5 , — + 3.6.5 2 3 . ti . y . 12.5-*

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

135. Ex. 3. Also, in the equation .r 3 — 12.r = 9, we haveJ2 h = 9, and ^(h 1 - + c‘) = 4 ; consequently b = -f, and c 1 —43 _ b‘ l = 64 — ^ == Ai 5 , which being greater than b 3 or s -/ 1this case belongs to the second class of series, or that of the

least roots. Now here x =^/(c+ 6) — iy(c — ^)=^/=——

^A/m-_9 _ 3/j j-i!4.37 8 - s/2-114378 = 2-2316619 -

A’2834950 = -9481669 = the root of the equationx 3 — S %/(b z — c 1 ) . x = 26, or ;r 3 + 3 3 /V . x — 9. And theterms of the two series being found as in Art. 113, namely,A+C + e + &c = -34051, and fe + D + F r|- &c = -03071,

also ^ being

36

V25P

, we shall have

By the 1st series

1-5321299

0-7082798

•34051 - lo;

36A/Z5 0

Series =- 1XB944I - - 0-2404097* = + 0-948167

— -791274 the least root.

By the latter series

- - 0-7082798

•03071 - log. 2-4872793361/350

= + -1568771 - - 1-1955596

= — -9481669

7912893 the same root.

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