446 Dr. hutton on Cubic "Equations 
133. Ex. 2. Again in the equation at 3 — gx = — io, 
we have 2b — - 10, and 3 -¥ b z + c z = 9; confequently 
b = - 5, and c z — 3 s - b 1 = 27 -25 = 2, which being 
lefs than b 1 or 25, this equation belongs to the firft clafs 
of feries, or that for the greateft root. Now 
x - ■</b J rC + \/'b~c-\/-^ + s / 2 + v / -5~v / a 
= — v / 5 — V 2 — v / 5 + 2 
- - ^'3*58578864 -^6*41421356 
= - 1*530600 — 1*858009 = - 3*388609 = the 
root of the equation x l — 3V 7 " b z - c z .x = 2 b, or 
^3 _ oj/ 21 ,x = — xo. And the terms of the two fe- 
ries are found as in Art. 1 10, namely 1 - 4 - — 8cc. 
3.6.9. 1 zbf 4 
Z 
— a— c— e - Sec. -‘9997359, and — 4^ + Sec. = b + d + See. 
3.6^ 
= *0089009. . AKo 4^4 = 4 v/ = 5 = - 4V 7 5 = -^320. 
Then ' . • T - " c\- 
By the ift feries 
By the 2d feries 
feries 
X 
*9997359 - log. 1-9998854 
— ^320- *- - 0*8350500 
~ - 6*838098 - 0*8349354 
= + 3*38860 9 
*0089009 - log. 3*9494339 
• — ^320 - - 0*8350500 
feries =7 *060881 - 2*7844839 
X 2= — 3*388609 
— 3*449489 the greateft root 
— 3*449490 the fame root 
And thefe values of the greateft root are nearly the fame 
with that found in Art. iio. 
134. But 
