j-6 Dr. Waring on the 
Ex. 9 . Let the feries be of the fame formula a + 6x + cx* + dx -f* 
&c. and in it for x be fubftituted p, -p, $p, - 3 ^, 5 ^, - 5 /, yp> 
— jp, . . . . np , — ^ and ^ ; and the fums refulting be S r , S“ r , S 3 , 
S - *^ S$, S -5, S 7 , S~ 7 , . . . . S% S~”, and S w ; then will 
1 2 
— I . n z — 9 . w * 2 — 25 . w 2 — 49 . . . * n z — n — 2 X m + n 
n z ~ i . rf — 9 . w 2 — 25 . « 2 — 49 .... « 2 — »~2 2 x 2n~2* Xi.2.3.4 • 5 •• n 
xS’-Ax 212* x S - ” — - x B x - CT + ” - x S*-» + C x — 2 ±? x 
W + » I w — « -j“ 2 ;« -f — 2 
S~”+* + *.Tl X D X X S”-4 -Ex 2Zl±i s-”+4 - — 2 F 
2 w — «+4 m + n — 4 3 
[x m + a .-4 S-t + ^ X G X S-+ ( + - 3 X H X X 
m — n + o m + n — 6 4 th — n + # 
S«— 8 -& c . nearly, where the letters A, B, C, D, E, &c. 
denote the preceding co-efficients of the terms S*, S"" 7 *, S”"~ 2 , 
S~ *+% S’ 2-4 , S"~”+ 4 , S”~~ 6 , &c. refpedtively. The co-efficients 
of the terms S 2W ~‘ 2f and S ^^ 25 will be M = - — Ltd. x L x 
s 
m + n — 2 s + 2 an ^ N-M x m ~ n +21 ; where L, M, and N denote 
m — n + 2 s m + n— 25 
the co-efficients of the terms immediately preceding each other, 
that is, of the terms S -»+**— '-^ S' 7- " 2 ', and S“"”+ 21 . The fign 
of the firft co-efficient M will be -f or — , according as s is 
even or odd ; the fecond term N will have a contrary fign to 
the firft. 
Thefe feries may be made to begin from any term, which 
may be eafily found by the method of correfpondent values, 
and the fubfequent terms from it by the given law ; its pre- 
ceding terms may be deduced from the fame law reverfed, that 
is, by putting the numerators of the fractions multiplied into 
it for the denominators, and the denominators for the nume- 
rators. 
2 
From 
