Infinite Series 
89 
!4.LetA x 2 4- /3 . z + y . % 4- X £ 4 e X See . +Bx?-h«XHy « 
# 4 - s . 2 + 6 • &c. 4 Cxz 4 a . £ 4- fi X ^ 4- ^ • % 4 e . &c. 4- D X 
% 4 ^ • # 4 / 3 .2:4^x242 • &c. 4 E x # 4 « • ^ 4 /3 • % 4 y ^ 4 ~ 
X&c.m, whatever may be the value of 5 ;; then will 
An 
D = 
0 — c& , y — a. . <i) N — a . &C. 
,B = 
a-i3 . y-£ i # . &C, 
,C 
*■— y . 0-y . ^-y . &c. 
,~-<K /3 — ^„-.y— &c. 
■, &c. 
1 5. Let the general term of a feries be L X <?% where e is lefs 
than 1 ; the fum of the feries can be expreffed in finite 
terms, when the above-mentioned quantities — 
+ 
7 
/* p*+i pot 4- 2 
/* + 3 
1 
• . 4" 
1 e 
o * — - 4* ttt 4 
4 
/«4« ? ' p @ 43 
.4 
7" 
$y i>y+i^y+2 
1 
1 
/>y4^ 9 /> £ 1 />$+ I ^^4-2 3 
p@-\rm ^ 
4-rrr-... 
» o, &c. n o. This never happens unlefs ^i, 
16. If the general term be any rational fundHon of % into the 
exponential e* 9 viz. ■ *»-,» < 
az^bz *4 cJ 2 4&c. 
Z + cc XZ + K + b XZ+« + ^ X.&C. X24-/3 X 
x e* ® KL 
»4 -@ + b' n xz+&+k' n K&c.Xz + / xz+y4-^" r ^2;4y4-i' /r x&c. 
where h 9 h', h" 9 See. k 9 k\ k'\ See. /, m 9 m' r m f \ See. n 9 n' 9 n'\ 
See. r, r', r'\ Sc c. & c» denote whole numbers, and neither 
Von. LXXVL N 
