and Infinite Series. 425 
and the only real root, or the firft feries, becomes 
2.4/ b x : ~ + &c. which is the fame root as was 
3.0.9 
before found in Art. 73. So that in this 1 6th cafe, both 
this feries and the feries in Art. 67. converge, and give 
the fame and only real root. 
95. When a becomes negative, then c z becomes ne- 
gative, and the roots all real. But in this cafe the feries 
only begins to converge when - al — 2 b\ for then - c~ 
becomes = b\ and then, making the proper change in 
the figns of the terms, the three roots become 
1 ft. - 2 4/b x - ~ 5 - + -^• 8 ’ 11 ■ 8cc. the leaft 
3 3 ; 6 . 9 3.6.9.I2.IS 
root, and 
{ 
+ 4/ b x 
3 3-6.9 
2.5.8 .II 
3 . 6 . 9 . 12 . 15 
± 4 /b V3 x : 1 + — 
° 3.6 
Sec. 
2 . 5. 8 . II . 14 
&C 3 
3.6.9.12 3.6.9.12.15.18 
the two greater roots. 
96. I have here faid, that the firft of thefe three roots- 
is the leaft of them.. To prove which, I aflert, that 
v / 3x — ■ 2 ' 5,8 - . Sec. is greater than 3 times 
- - ■ 2 • 5 - See. for 3 x : — _ Sec , = 1 - ~- 5 8cc. 
3 3 • 6 • 9 3 3 - 6-9 6 -9 
is lefs than r, whereas 1 + Sec. is greater than 1. 
Confequently, the lefs of the two latter roots, namely, 
4 /b 
