420 Dr. hutton on Cubic Equations 
80. When a 3 is greater than b 2 , c 2 will be negative, 
and then, changing the ligns of the odd powers of c 2 f 
the three general feries will give the three roots of the 
equation, which will always be all real. In this clafs are 
two cafes, namely, when c z is lefs than b 2 , and when 
they are equal, which is the limit; for when c 2 becomes 
greater than b 2 , the feries diverge. 
81. Now when a 3 is between b 2 and 2 b 2 , then <ris 
negative and lefs than £*, and the general feries give all 
the three real roots by changing the fign of every other, 
term. 
82. And when a 3 = zb 2 , then — c 2 — b 2 , and the three 
roots become thus : 
8 cc. the 
3.0.9»I2 3.6.9.12.15*10 
firft or greatefl root, 
tyb x M + 
g 4/6 x : 1 + — — ; — — , 
3.6 3.6.9.12 3.6.9.12.15.18 
2 
iP> 
5-8 
2. 5. 8. ji . 14 
&C. 
^ 1 3.6.9.12 3.6.9.12.15.18 
±J/b x x - - 2 -ti + 2 • 1 • 8 _ L 1 — See. 
3 3 • 6 • 9 3.6.9.12.15 
the two lefs roots. 
83. The firft of thefe 3 is the greateft root, be- 
caufe 3 /b x : 1 + ~ See. is greater than 
3.6 3.6.9.12 * 
v/£ x nA x : - — See. for 1 + -^-r &c. is greater 
than 1, and ^3x1-- 
3.6 
- 2 - / 5 ■ See. =Vjx ; 1 
3.6,9 3 
^i&c. 
6.9 
is. 
