426 Dr. hutton on Cubic Equations 
■s/b . v^3 x : i + 2 — See. — s/b x. : —r— Sec. is 
J 3.6 33.6.9 
greater than the firft root 2 ^/b x : — - 2 ' 5 .. See. That 
° 3 3 • 6 • 9 
is to fay, here the firft is the leaft of the three roots* 
while in the other clafs of feries the firft is the greateft 
root. 
97. Hence, comparing the value of any one of the 
roots here found, with the value of the fame root as 
found in Art. 8 2,we obtain the relation between the two 
feries that are concerned in them, namely, that the feries 
1 + “ —-- v 5 ' 8 ~ 5 ' 8 ' 1 1 ‘ I4 ~g & c * * s to t ^ ie feries 
3.6 3.6.9.I2 3.6.9.I2.15.18 
J __ 2 - 5 - 8 .ii. 14.17 , &c . as 
3 3.6.9 3.6.9.12.15 3.6.9. 12. 15. iv. 21 
3 + 1 is to V 1 , or as 2 + V 3 to 1, or as 1 to 
2-^3, which are all equal to the fame ratio. And the 
fame thing appears from Art. 85. 
98. When - a 1 becomes greater than 2^% — c r is 
greater than b~, and, by the proper change in the figns,. 
the feries for the roots in all cafes of this kind become 
1 ft x : i - + i ea ft root. 
Vr 3 3.6.9c* 3.6.9.12.15c 4 
and < 
.. ^ X 2 . „ 2.5.8.H 
+ x : — — , + - — 5- 
b 
tU 
4 &c.V he two 
3 3-6.9^ 3 .'6 . 9 . 12 . 15c 4 i greater 
*yc .\/ 3 x : 1 + — _ .. 2 - 5 - 8 iL See. f roots, 
L 3 3. 6 c z 3,6.9. 1 2c 4 J 
99. Let us now illuftrate all the foregoing feries for 
the roots of cubic equations, by finding by means of 
them 
