182 
Dr. Waring on the 
(x 
Cor . Let B =:o, then the feries D^‘=:A x 
K**s+ < r +*’>'- s+— P+*v x , + &c .), 
* • 2 . 3*4*5 3. 4. 5. 6. 7 ' 
3 
and the feries i = i + P^4 
P 7 + A 2 
* 2 4 
(P*4-A 2 ) 2 x , , p (P* + AT , (F a + A*)3 
i. 2*2*4 x»2»3*.*5 i • 2 • 3 * • 
X 
■jr\ P 2 -f A 2 
Px * -M ~ ~ .v , , 
1.2.3 1 . 2 . 3 . 4 ‘ 1. 2. 3. ..5 '1.2.3. .6 
n 
4&c. ; the co-efficient of the term will be (P' + A 2 )^ or P 
n— 1 
jx (P 2 + A 2 ) * , according as n is even or odd. 
If in the equations before given for a be fubflituted a = b 
inftead of n-\- b 9 then in the other quantities for b fubflitute 
— b. 
3. If in Cafe 2. the difference between the two quantities 
(1 4- P^ + Q^ 2 +'&c.) x (1 +P^-f Q^ 2 +&c.) and (Atf+Btf 2 q-Of 
4-&c.) x (A£q-Bff 4-C^ 2 4- &c.) isaffumed — r 4-P x a + b + Qx 
a + b 4-&c., then in the feriefes before given for A, B, C, &c* 
write refpe&ively s/ — lA, s/ — iB, \/ — iC, &c., and there 
will refult the correfponding feriefes. 
The fame principles may be applied to many other cafes. 
4, Equations of thefe formulas may be ufeful, when the 
fums of the feriefes correfpondent to a value ( a ) of a 1 are 
given, and the fums of the feries correfpondent to a value 
{ a 4- b') of x is required, b having a fmall ratio to a : for inffance, 
let the given feries be x — 4 — p &c. ; the 
equation found in the firft cafe is a-\-b— ( ' a + h ~ 4 — — ^ 
^ 2.3 2 . 3 . 4 . 5 • 
&c. = (a — ~ — p — — — &c.) X (1 — — — 4 &c.) 
v 2.3 2. 3. 4. 5 y v 1.2 x.2. 3. 4 y 
, f a 1 d 3 * O \ /7 ^ ^ c v 
HP (1 — ~ — ~ 4 * 7 r — — $ccA. x f b — 4- 1 ™ &c. ) 5 
x.*, 1 .2.3.4 2.3 2. 3*4*5 
but 
