3 Z 
Dr. Waring on 
arz-b—, b — —- tna, See. 
I 
e — i ' e 
If e— i, then affume az^ 1 + &z* + &c. for the fum 
which rule was firft taught by M. J. Bernoulli. 
If the term be %™f h * for f b fubftitute e y and there refultfc 
the fame as before. 
1 8. Let P -= A + Bx n + Cx 2n + Dx*” + See . then will; 
the fum of the feries 
A 
B*" 
cc . 0 . y . £ . &c. a + « • 7+ w • & c * 
c* 
X P 
* + 2» . /3+ 2« . y + 2» • &c. 1 a/Jyi,&cc. 
-i- . x> &c. fx a p- ~ n . U— 
a J * 0 GC—@ .7—$. 
cc (3°— a y — oj 
■^-3 . &£C. X—fi fx^p - 
JL . _L. , . -i— . &c. x x—v fxvp — See. 
y a«*y @ — y ° — y J 1 
This may. be proved from the fubfequent arithmetical propo-* 
~ • I I I I O , I I I I o 
fition — ■ « - . - — •. t — - • +. - • — —q • — - ■ t" — - » & c. -K,- 
p - « y ~<x d— « p p 7 — @ o— - (d 
I 
a — 7 
I . 
9 @ . y • &ci ' 
— . &e. + i 
— y 
i i i 
y — ^ 
. • &C. 3 
1 9 . Let, the general term of the above-mentioned feries A + B> 
’ + fo 2i *+&c. be Hx %n ; then from the fums of the feries and 
the fluents of the fluxions x^p, xlp 9 x l p, &c. being given 
there follows the fum of a feries, whofe general term is 
az h +bz n T W” x YLx %n 
d+nz .'(3 + hZ:, 9 ..y+hz . . $+,«z . ,&c. 3 
number. 
where / denotes a whole 
T/ , TT m—l m - a w — 
If - . . ... * , or sa-m . — 
2 a z+i 2 
m — 2 
, where / is a whole number, and m 9 c/ 9 7/, & c * are 
m — Iz 
/z+ 1 
either whole numbers or fractions whofe denominator is 2, and 
a = a! n 9 /3 =■ (Z'n*, Sec. the fum of the above-mentioned feries can be 
found by finite terms, circular arcs and logarithms. If 
