TRACT 28.

AND INFINITE SERIES.

321

136. But, in Art. 62, the same root was found to behence then we shall have these first two following

equations, and by means of their sum and difference we ob-tain the other two:

3/(2ty7 + 36)- V'goyi- 36) 4- y '-I — 3 3 j0 rrl I , 3.5.8.11.31° „

72 V 3 U — 3 + 3.6.9.12,15.175* V *

• yf30V7 + 36)-y(2O, v /7-3G)-v'gl + 3 ,2. 5.81 & „

12 V /3 ' i0 ~ .3 . 6 . 9 . 175 + ’

M20V7 + 3S)-M20a/7-361 , „ _ I 2 • * ■ 81 , &c .

36 v' 350 — 3+3.6.9.n5 + CC ’

v/21-3, , 1 3.6.81

36

- 4/350 =- -

u . 175

3.5.3.11. 81*3.6.9.12.15. 175*

&C.

And the last of these agrees with one found in Art. 115.

137. Ex. 4. In the equation „r 3 — 15.r = 22, we have23 = 22, and */(b % + c 3 ) = .5 ; consequently 6=11, andC l =.5 3 — b’~= 125 — 121 =4; which being less than b'~, or121, this belongs to the first class of series, or that (or the

greatest root.

Now x = 3 /(b + c) -\-fy{b — c) =4/13 -f- 79 = 4‘4314186 =the root of the equation .r 3 — 34/(117) . x = 22. Andthe terms of the two series being found as in Art. 117,we have the first = a — c — &c=l — •0000150 =*9999550, and the second = B + D + &c = ‘0036731 -f-*0000008 =-0036739. Also 44/6 = 44/11 =4/704. Hence,

By the 1 st series

•9999550 - lug. T-9999S05

3/704 - - - 0-9491909

series =+ S-8956200 - 0-9491714

x =— 4-4314186

+ 4-4641014 greatest root.

By the 2J series

•0036739 - log. IFoGSISISV704 - - - - 0-9491909

series = + -0326827 - 2-5143182

x =+ 4-4314186

+ 4'4641013 the same root.

Which nearly agree with the same root found in Art. 117.

138. But in Art. 64, the same root was found to be1 + -V/12 = 1 +2t/3, hence we obtain these two first equa-tions-following, and their sum and difference give the othertwo :