[ 762 ] 
lues, fo refulting together, all marks of impojjibility 
will difappear. 
If, in the feries to be fummed, the alternate terms 
(viz. the 2d, 4th, 6th, &c.) Ihould be required to be 
taken under ligns contrary to what they have in the 
original feries given ; the reafoning and refult will be 
no-ways different ; only, inftead of making p l + q* 
+ r l (or p n -f- q n + r*, &c.) = + 3 (or + n), the 
fame quantity muff, here, be made = — 3 (or — n)* 
From whence, p* being = — 1, q n = — 1, &c. the 
values of p, q> r, &c. will, in this cafe, be the roots 
of the equation z n -f- 1 = o. 
It may be proper, now, to put down an example, 
or two, of the ufe and application of the general 
concluhons above derived. Firff, then, fuppofing 
x 2 
the feries, whofe fum is given, to be x -j- {- 
.v 
AT* 
— 
3 4 
x m + n 
+ ~r~z. + 
X 
X m + 
4 - U 
m m-\- 1 
x 
m + n + 1 
m -J- n m-\- n-\- 1 
— }-, &c. 
x m + 2 
+ — 
= — H. Log. 
1 — x(=S)-, let it be required, from hence, to 
lind the fum of the feries ( j J 
\m m~\-n m -J- 2 n 
tcc.) arifing by taking every term thereof, be- 
ginning with that whofe exponent (m) is any integer 
x m 
lefs than n. Here, the terms preceding being 
tranfpofed, and the whole equation divided by x m y 
we 
