TRACT 28.

AND INFINITE SERIES.

115

equation x 3 — 3^/(i 2 —c 2 ) . x = 2b. Now make 2i 'yb — 1, andf 2

— — g 1 •, so shall the above become {^>(1 g) + ^/(1 — g)

_ , 2g’ 2.5.8ff4 „ , - ,

— i — r— ; — - „ „ — &c = the greatest root or the equa-

3.63.6.9.12 ° 1

• 1 c 2 . ♦

tion x 3 — — g 2 ) . x — And when g 2 or is negative,

these become

i'Ai+gs-o + i-gy-D = +

&c z: the greatest root ^jjTthe equation -jrg*)#

— x' that, in general, the infinite series

_ 2g’ 2.5.8^4 _ 2.5 . 8 . 11 . 14.?6

"*■ 3 . 6 — •

&C, is =

* " r " 3.6 3.6.9.12 3.6.9.12. 15 7 18

iV{4 + g•/ ± 1) + il/(l ~ g*S ± 1 ) — the greatest root ofthe equation x 3 — l qr g"-) x — Where the upperand under signs respectively correspond to each other.

123. Again,

... . 7\ , . 1 \ 25 1 2.54’ , 2.5.8.1144

V( + ) y'( t ^ 3/ C 2 X ‘ 3+3 . 6 , 9 c j + 3.6 . 9 . 12 , is c 4

is the least root of the equation x 3 + 3 3 /(c 2 —A 2 ) x — 2b.

this becomes

Then, by taking ~ = l, and-J-

V(i+g)-V(i-g) _ J, 2 Jig* „

2 g 3 + 3.6.9 SC

g:

the least root of the

equation

x 3 + —^r- ] And when g z or c 2

gative, this becomesMi+gV-il-Mi-gy'-OW-i

root of the equation x 3 ■

4S'

is ne-

2.5 g>

3 3.6.9

SVO+g 2 ) r = Z}4g’

+ &c = the least

x = 5T S - So that, in ge-

neral, the infinite series

JL ± 2 ■ 5 g 2 . 2-5-8 -Hg 4 + .

3 3.6.9 "t” 3.6.9. 12.15 —

2.5.8 . 11 . 14.17^63.6.9 .T2TT5 ."TsTsil

&C,

_ V(l+gV'±’)-V(l-5v'±l> .

ls > — 2gV±I- — tile Jeast root °* the

equation, x 3 ±

31/(i=R 2 )

4jt’

X

±1

4g 2 ‘