922 Lxtenfton of Cardan’s Rule to the 
, , , - . ~ . r . i aTb a^b 7 ta%b l 10 alb 4 
be equal to the infinite fenes cp — 7 — %— 3 - 
a 'xa q a 8ia s 2 /X.xar 
ziaiP i Z4# J b 6 26 iSaib 7 
1 $4a?b 6 
729 6561 a b 
h b z 
3 a 9 a 01 a” 243 ar 
- 8cc. or to cf x the infinite feries 
^ 3 a 9 a z 8 i< 3 3 
1 37?7 g i a> 
$b z ioZ> 4 2 23 s i54^ 6 2618b 7 
243<? 4 729 a ' 6561 a 6 137,781 £ 7 
I ..v • £ r • iA3 5C3 3 8D3 4 
the infinite leries i- ^ — — 
6 a z 
3 a 
9 a J 
1 2a 4 
— See. or 
1 iEP 
15a 5 
i 4 F £ 6 1 yGi 7 
18 a* 21 a 7 
&c. in which feries the numeral coefficients 
of the feveral terms are the fame as in the feries that 
exprefles the cube-root of a + b, but the terms which 
involve the odd powers of b (which in that feries are 
marked with the fign + , or all added to the firft term,) 
are in this latter feries marked with the fign and are 
all to be fubtraefted from the firft term, as well as the 
terms which involve the even powers of which are 
to be fubtraefted from the firft term in both feriefes. 
PROBLEM. 
23. Let it now be required to refolve the firft cafe of the 
cubick equation x 3 - qx = r, in which r is greater than 
— j, or L is greater than L, by means of an infinite 
feries derived from the expreffions given by cardan’s 
rule. 
SOLUTION. 
We have feen in Art. 1 1. that, if ss be put =- - 
4 27 
the value of x in this equation will be = \J'yj+s + 
V 3 
