f 
Method of correfpondetii Values , See, 
___ __ 1 7 S 
g — m ' 1 * « ^ 2 — 9 . — 16 . t . {m 2 —(n — i) 2 ) * 
i. 2 . 3 . 4 . 5 ...( 2 «— 1) ~~ x 
y 2” “I x p> x + .. o„_ T . ^ m-~n ~- 7 
1 
m — n — 1 
x S"- 1 + C X 
OT-i-B — 2 
S—>+2 4. 2 X D X 7 ” + ”_J X S"- 2 - &c. nearly, when A, B, C, 
77 L — « — 2 
&c. denote as before the preceding co-efficients. The coeffici- 
ents of the terms and S” - " 1 will be refpeftively L x 
- — - — — L and x M x - , L and M denoting the pre- 
ceding co-efficients, which are to be taken negatively or affir- 
matively, as s is an even or an odd number. In this feries when 
x = o, the correfpondent fum = o. 
Ex. 7. Let the given feries be of the formula a + bx + cx z + 
dx 3 + &c. ; and in it for x be fubftituted 0 , p 9 — p , 2 />, — ip , %p y 
— 3p . . . np 9 -pp and mp, and for the fums of the refulting fe~ 
riefes be wrote as before S% S l , S"~ x , S% S"“% . . . S% S"“% and 
m . m z — 1 . w 2 — 4 . T7/ 2 '— 9 . . . (m 2 — n~ i “) X m — n 
S m ; then will S/72 = 
n . r? — 1 . 71 — 4 . r 2 — 9 . . . {nS — n ~i‘)X2B 
X S“ a + Ax — — x S+" - — x B x -7 — — S“ K + X - &c. this feries 
I m-\-n — I 
m — n 
obferves the fame law as the feries given in Ex. 5. and only 
differs from it by the laft: term So not vanifhing, that is, 
being = o. 
Ex. 8. Let the feries be of the preceding formula a + bx + 
cx‘ ll +dx 3 + &c., and in it for x be fubftituted o; p 9 -p; 2 p 9 
- - - 
— 2 p ; 3/), — 2 )p > • • • « — * p 9 -n—ip, np, and mp , and the 
fums refulting be So, Si, S~~ T , S% S~ 2 , . . S”^ 1 , S” and S m ; 
m—n 
y • 11 o m . m — 1 • m ~~ 4 • „ m — (n — I ) c „ * 
then will S/72 ~ — v -S"— A x * x 
I . 2 . 3 . 4 . . • 2* — I ?n n l 
— &c. the fame feries as in Ex. 6, and differs from it onlv 
•J 
by the laft term So not vanifhing. 
E e 2 Ex, 
r 
