174 Dr. Waring on the 
Ex. 4. Let the feries be of the formula A + ax 1 -f- ix A + ex' + 
&c. ; then will Sn = n ~ 1 “ ~~ ~ T ~ r 2 *~” + 
h— 2 
— x zri — 2 b« — I x 2^ — I . 
». n 
in — 2 in — 60 
s« - 3 - 
«- 4 
X 2U — i . 
n 
x S« — 2 + - — - x zn — 1 . 
« 2 
2 LH^ . ?” - Z J x — — - S;z — 4 + &c. nearly, of which the general 
term is - — 1 x zn — 1 . 
n 
2 n — 2 in — 3 
1)1—1+ 1 2/2 — 2/ c ; , 
. . • . — X 7 X b/i — /. 
3 3 * l 
Ex. 5. Let the given feries be of the formula ax 4- bx~ 4- cx 3 
+ &c., and in it for x be fubfhtuted p, - p, 2 p, - 2 p, ^p, - $p 9 
. . . , np , — np and mp, and for the fums of the relulting fe- 
riefes be wrote relpetlively S 1 , S — *, S 2 , S -2 , S 3 , S“ 3 , . . . S% 
and S in ; then will Sm = 
m . rn — 1 . m — 4 . m — q . m — 16 . . . m — // — i • w — n * 
n . 22* — I . n 1 — 4 . n z — 9 . n z — 16 . . . n ' — n— 1 X 2 « = i . 2 . 3 . 4 . . in 
x S~" 4 - A X 
” + »C4., 2 « „ T> „ m ~ n g-a+i _ Q y » +"- 1 
m—n 
S"- r + 
2/2 — I 
x D x 
I 
m — n — 
- 1 
m + n— I 
X 
m — n — I 
m + n — l 
X 
S _„ +2 + *j^ xES ,_ 2 _*-2 x . 
m — n — 2 
F x 
m — n — 2 
g n _|_ 3 __G x 0 x S” 3 + &c. nearly, where 
m—n — 3 
the letters A, B, C, D, &c. refpedlively denote the preceding 
co efficients. In general, the co-efficients of the terms 
and S*— 5 will be refpe&ively M= x L x ~ n ~ s ~ tJ. and 
s m+n—s 
M x 
m+n — s 
m— n— s 
, where the letters L and M refpedtively denote their 
preceding co-efficients ; the co-efficients are to be taken affirma- 
tively, or negatively, according as s is an even or odd number. 
Ex e 6. If for x in the preceding feries be fubftituted p, -p 9 
2 p 9 - 2p f - $p, • * * n— ip 9 —n - ip, np refpedtively, then 
Sm 
