8 8 Appendix to a Method of extending 
is equal to the fum of the two following feriefes, to wit, 
2 a — - ~ - - &c. (in which all the terms fol- 
lowing the firft term are marked with the fign — , or are 
fubtradted from the firft term), and A ~ + ^-,+ &c ; 
- &c. is equal to 
and the feries 2 a 
zC- 
2 Ez 4 2 Gz ( 
the fum of the two feriefes a + — — — + E? — ^ + 
e es e e + 
•TV Gz 6 o 1 Bz Czz Da 3 Ez 4 Fz 5 Gz 6 
e 3 e° e ee e e -e e 
Sec. which are refpecftively equal to the cube-roots of the 
binomial quantities i + — and i — - . Therefore the 
x e e 
- . zCzZ 2Ez 4 2Gz 6 o • /of " % /-if 
feries 2 A+ — — +s~~ &c. is = v 3 i +— ■ + \/ 3 i — 
ee.ee I e I 
the infinite feries A ~ + + See. Confequently the 
expreflion \/ l e x the feries 2A + 
2CZZ 2 Ez 4 
+ 
- Sec. 
is — *J 3 e x \Z 3 li +-+\/ 3 ex\/ i [i — -) + V 3 e x the infinite 
\ /> p ' 
feries 
A.CzZ 4 -Gz 
+ 
&c. = \/ 3 j e+z + s / 3 e — is] + s/ l e x the 
infinite feries 
a.Czz . a.Gz ( 
+ ^7T 
See. = \/ 3 e + #1 + \/ 3 e - 
z + 
/, . r . Czz Gz 6 Lz 10 Pz 14 Tz 18 0 
av% x the feries — + — j- + -n- + — — + — 8 -+&c. There- 
“ ee e e e 4 e 
fore tlie root of the equation x 3 -qx = r, in the fecond 
cafe of it, in which ~ is lefs than is equal to 
\Z 3 fe + «] + \/ 3 \e - ~z ] ' + 4\/ } e x the feries — + ~ + ^5- + 
^ + 8ec ; of which expreflion the firft part, to 
wit, 
