uyo Dr. Waring on the 
4. In the fame Book is aflumed a feries ax' + Z.v'+' + cx r + 1 ‘ + 
dxr+v + &cc. of a more general formula than the preceding, 
and in it for x fubftituted <x, (Z, y, $, &c., m ; and S«, S/ 3 , S^, 
SS, &c ; S#z for the refulting fums, and thence deduced 
m X m — $ . in — y *771 — 0 . &c. 
/x« 5 -i 3 ! . a * — y* » ^ 
X Saj-f 
111 X 771 s — a* . ffi —y .m — t • &c. 
/3 r x & — a 5 . (3 s — y s . I s . &c. 
xS£-f 
x — a,* • m 
@ S *771 — / . &C. 772 r x m s ~ m* — (3 s . m — y X &c. 
- X Sy + — — 1 
" x y J — u . y s — $ * y — ^ • &c, ^ X $ s — &* . — /3 1 . d' J — . &c, 
x SS+&C. nearly. 
Cor. If for r and s be affumed refpe&ively i, the feries be- 
comes ax+bx z + cx 3 - f&c. of the fame formula as the pre- 
ceding: if r = o and s= 1, the feries becomes a + bx + cx* + 
&c. The latter cafe will be the fame as the former, when one 
of the quantities (a) fubftituted for x and its correfpondent 
fum S«, both become = o, and the equation deduced in both 
cafes the fame. 
5. If 7T, p, cr, &c. refpectively denote r, r+p, r + zp 9 . . • 
r-\-n — 2p, r + n—ip , and T~r + np; and S, Si, S2, S3, . . . 
S/z— 2, S«— 1, be the fums either refulting from the feries 
ax 4- bx z q- cx 3 q- &c. or the feries A + ax + bx 1 q- cx 3 q- &c., which 
refpe&ively correfpond to the values r 9 r+p, r+2p , &c. of 
x ; and Stf the fum of the fame feries which correfponds to the 
value r+np of x\ then will Sn=zn§n—‘i - n * n -^-Sn - 
— . _ bn - 2 - ... ±n * 02= nbi^zb nearly ; this 
23. 2 ^ ' 
equation differs from the preceding by the laft term S not va- 
niftung ; in the preceding cafe S became = o, for it was the 
fum of the feries ax + bx z + cx 3 q- &c v which correfponded to 
Q* 
6. From 
