Second Cafe of the Cubick Equation x % -qx~r. 923 
^ f~- J. For the fake of avoiding fractions, let e be 
put =— . And we flhall have x- s/ 3 (e+s +s/ 3 \e-s. But 
(by Art. 21.) </ 3 | e+s is = x the infinite feries 1 + -i_ 
y 
&c ; and, by 
SS £J 3 IOJ 4 2 2S S I £ 4 / 26l8i 7 
243 ^ J2ge 5 6$bie 6 
9 ee 8 it? 3 
1 37 > 7 8lf7 
Art. 22. \/ 3 j e—s is = e* x the infinite feries 1 — L _ !L _ 
%e yee 
1 or 
S 4 * c 
2618/ 
03 4 5 4.6 „ — 5— 8cc. Therefore \/ } \e+s + 
Sie 2 43^ 729^ 6561^ 137,781^ ! J ^ 
v/ 3 f 7 - j is equal to e ij x the fum of thefe two feriefes, 
that is, to e j x the infinite feries 1 — — - ?o8 f — &c: 
and confequently the root of the equation ad - ya; = r is 
= e> x the infinite feries 2 
2SS 2 or 
308/ 
9^ 243^ 6561^ 
— &c. ad infi- 
nitum. Q. E. F. 
24. Note. This feries muft always converge, becaufe 
ss, or - - — , is always lefs than — , or ee. And, when 
4 27 4 
w is confiderably lefs than ee, or - - 1 — is confiderably 
lefs than or ^ is very little greater than q ~, the con- 
vergency of the terms of this feries will be fufficient to 
make it ufeful. But in other cafes, when - is much 
4 
greater than q —, (as when it is triple, quadruple or quin- 
27 
tuple of it, or ftill greater,) the terms of this feries will 
converge fo flowly as to render it very unfit for practice. 
And indeed in the molt favourable cafes it will, as 1 be- 
lieve, be lefs convenient in practice than the expreffion 
5Y 2 
