88 Dr. Waring on 
fulting be multiplied together, and their relpe&ive contents be 
called /“ +3 , />' 1+4 , . . ; then, if the fum of the feries 
can be found, will — H — H 1 — 1 — . . . — ~ —o. 
pa, pcc-f-I 2 p* m T 3 pct-f-n 
In the fame manner fubtraft / 3 , / 3 -f i , 0 + 2, . . 0 -bm re- 
fpe£tively from all the remaining quantities, and multiply 
their refpedtive remainders into each other, and call their con- 
tents refpe&ively f^ 9 f ^ +I 9 then will Lq 
p& 1 
= 0. 
p0+2 p@ + m 
Subtract 7, 7+ 1, 7+2, . • 7 + r refpe&ively from all the 
remaining quantities, and multiply their refpe&ive remainders 
into each other, and call their contents refpe&ively p?, p y +\ 
^v+2 . . £?+ r ; then will -h-j ^ — | — . . — L_ = o. 
* 7 r py py+i py-\r 2 py-\-r 
Subtrad S, S+ 1, 2, . * s from all the remaining quan- 
tities, and multiply their refpedHve remainders into each other, 
and call their contents refpe&ively p\ then 
wIU p + jki +^r, = 0 ; and fo on * and > vice ver / d > 
if the fum of the above-mentioned fra&ions be refpeftively = o ; 
then the fum of the feries, whofe general term is the given 
one, can be found ; otherwife not. 
13. If the fum of the feries, whofe general term is 
.. x 2 ; 4-/3 ..Z + /3 + /WX z+y . %4-y + i , . 
— rj, can be found in finite 
-fy-fr X5Z + ^ . Z + o4- I . . Z + $+ S X &C.~ H 
terms ; then the fum of a feries, whofe general term is 
tl (where / denotes an affirmative number) 
can be alfo expreffed in finite terms. 
14. Let 
