Method of correfpondent Values, &c. i b i 
this feries is, firft, that every third term vanifhes ; and, fe- 
condly, the figns of every two fucceffive terms change alter- 
nately from 4- to - and — to + ; and, laftly, the co-efficient 
; and the feries i 4- qx -f rx T 
2 3 B 3 7 2 3 B 4 
of the term is 
n — t 
X 
4- &c. becomes i 4- - x — 
24B5 
n pj 1 — 2- 
2B* 
+ 
I • 2 • 3 
B 
- x — 
A i . 2A 2 
X 1 - 
-zX + 
I . 2 . 3 A 3 
■x + 
X 
1.2.2. 4A. 4 
i . 2 . 3 . 4 . 5A 5 " ‘ i . 2 . 3 . 4 . 5 . 6A 6 ‘1.2.3.. 7 A ' 
&c. In this feries the figns of three fucceffive terms alter- 
nately change from 4- to — and — to + ; and the co-efficient 
ft T> n B — I . r >« 
- , • 2 X B _ 2 X B j. 
of the term x” is — — or according as n is 
I • 2 • 2 . . nA ! 
I .2.3. »A” 
divifible by 3 or not. 
2 . Let a fenes 1 4~ P^ 4~ Qx 4"R^4~Sx 4 4^T^4~&c» be of 
fuch a formula, that if in it for x be fubftituted a + 6 , there 
refults a feries i4 -Px^ 4 -^ 4 -Qx^ + ^ 4 Rx^f/ 5 3 + Sx^4^ 
4-&c.rr(r4-P^4- Qf 4- R^ 3 + S0 4 -f - &c. ) x ( 1 +P b-\- Qb z - J-R3 : 5 
-f-S^ 4 +&c.) -^(A^-f-B^+C^-f-D^+^c.) x (A£+B£ 2 + C£ s 
-j-D^ 4 +& c *)> then will the feries Ax4-B# 2 -b^ 3 + D;r 4 --f- 
&c.= + -+Ax^|^)x 5 -1 2 B 3 — 2PAb*+a 2 x 
6 A 2 
— ^ 4 -f &c., and the feries I +P^+Q^ 2 ^R^ 3 -|-&c. sz 1 4- 
a: 4 -f- &c. Let 
■P , A 2 + P 2 * . aAB -f P xA 2 +P 2 3 , 4B 2 + A 2, 4- P 
irx 4 - x ffi- — z" x ■¥ 
6 ‘* 24 
¥Lx n ^ 2 -f- Lx n -~ I Ma?” be fucceffive terms of the feries Ax-\-Bx z 4- 
Cx 3 -{-&c., and ¥dx n — z 4-L / ^' J "” I -f-M / x” fucceffive terms of the 
feries i -hP^+Q^ z +R^ 3 + ^ c * 5 then will A x L-f-P X L 7 = # 
X M 7 and B x K+Q X K 7 = n . x M 7 exprefs the law of the 
feriefes. 
V ol • J.XXIX* F f Con 
