396 | : Dr. W Aarne. on. the 
on Ox bye (+ &e. 
yete.rztrte.7e+2r+e. 
number (N) of factors in the denominator diminithed by unity.. 
Byom a, 25.75. SG 7 and ¢ being § given, eafily can be acquired by 
fimple equations, or known theorems, the required co-efficients . 
a, b;.¢,,8c.. lf m=N —1 and « and a+-b+c+d+&c.=0,. 
then the fum of the feries refulting will be finite,. é 
. 
Ro 3 where m is not greater than the: q 
° T , é 
3. If ‘the terms. of the fenes aftumeéd, = ee 7 ae 
NE a RS 
+ &c.. be alternately affirmative and negative; then. 

ae Es. M=32 : 
vathe: preceding) cafe find)" * "23 See a 
by. the preceding ca dis Sp ee 
b TT 1 4 
: + +&c, Where the terms of the refult- - 
robe repete rze+arhe 
ing feries are alternately. affirmative and nee let the - 
Be es a Tt ane op Seed . 
TELE + Bb +E o a rey, 

two fabfequent terms be fuppofed - 



ee 1 RE. and amt E et ee ere: * a Sec, 
is rh re4 re Tate. Ta Ta2r WHT. 5 ay 
b 
— Ft & ee. of which+the one is affirmative-and the 


r2-+r-—-te | 2+ 2r-Fe 
other Sitios : reduce the refulting feries to-an affirmative one 
by fubtracting the fubfequent term from its preceding, and it be-. 
, m B Sri. The ate 
(rz -tur-pey (we Ox" +&c.)—(rz +e) (ext $RetE ke.) 
rzFe. r2+7+e.7e parte... -7S+rn-+e - 
Mm——T 
comes 


™ - 
N— Mam + &e. be y e 
ps, ae ee Maal tne ies 9 eS sap " 
Tree. v2 p-rbe.refrate rZ-+e * aaa + ee In this 
cafe, fince,two terms are added into one, the diftance from,the 
ws - a z x 3 
firft term of the feries will be —,..which fuppofe = w; and.write 
2 
2w for % in ‘the above-mentioned term, and: there refults 


n— mraz +-&e. n—mra% 2” w+ &e, 
tehe . rztrte.e  1Btmrte. wwe « 2wtrte... 27w+nurte 

a 
2rw He 

