TRACT 28,

AND INFINITE SERIES.

119

181. But the same root has been found to be — 2 +V3,in Alt. 59 ; and hence we obtain the sums of these two par-ticular series, thus :

yi3-y° +2-^/38

l/i3-y9-2+y 3 38

1/121 = -r +

. 5 . 8 . 11 . 2 * . 0

mil + &c >

3 1 3 . 6 . 9 . 12 . 15 .

a loi — 2 • 5 ■&_ , 2.5.8.11 ^14 . 17_. 2 6 __ „

Vl-1 — 3 _ 6 9 ; n i +3 .6.9.12.15 .'l8'.31 T ll*® 10.

132. Also, by taking the sum and difference of these two,we have

idlr^ yi21 = i +;

5-^121

2.5.2 33 " 6~. 9 . 11*

2 . 5 . 3 2.3 . o . a . 11-

+

2.5 . 8 . 11 . 243 . 6.9 . 12 . 15 . 114

_3 ■ 5.8 . 11 ■ 24

+ 3.6. 9TlS .15 . 11

+ &c, and— &c.

And this last expression agrees with what was found in Art.108.

133. Ex. 2. Again, in the equation .r 3 — 9.r = — 10, wehave 2b = — 10, and 3 3 /{b' L + c 2 ) = 9 ; consequently b= — 5 ,and c 1 = 3 3 — = 27 — 25 = 2 ; which being less than bf-

or 25, this equation belongs to the first class of series,or that for the greatest root. Nowx = V {b + c) + V{b - c) =*/( - 5 + V2) + M -5 - V2)

= ~\/{5 — <Z2) -MS + V2)

= — J/3’58578864 — $/3'4142 1356

= -1 530600—1-858009 = — 3-388609 = the root of theequation x 3 — 3 ^(b 1 — c z ) . x = 2b, or x 3 —31/(21). x = — 10.And the terms of the two series are found as in Art. 110,

namely 1 — 3 ^ 5 g - &c = A - c — E — &c ='9997359,a nd - + &<i=b + d + &c = -0089009. Also 4 %/b =

4%/ — 5 = — 43/5 = -3/320. Then

By the 1st series

•9997359 log.~9998854-y 320 - - - 0-8350500

series =— 6-838098 - 0-8349354

x =+ 3-388609

By the 2d series

•0089009 log. ~3-9494339-%/ 320 - - - 0-8350500

series = - -060881 - - 2-7844839x =- 3-388609

— 3-449490 the same root.

— 3-449489 the greatest root.