4° - M r. v ince’s new Method of 
whofe rta difference is rro; fubflitute, therefore, this feries of 
quantities for refpe&ively, and the given feries becomes 
afi-tib+ n . 7 1 -f rn . c 4- Sec. ( a + n 4- m . b -f // -j- m . n + z m . c + &:c. 
= = 1 ■ - ■ 
n 4- rrn 71+771 ~ 
n 4- r 4- l . fti 
^ 4 * n + 2 /« • b 4 - « 4 " . « 4 ~ Vn ■ f-h Sec 
= ^ : — ' 4- &c. 
A 4- 2 m . . . . n + r+z . m 
which manileftly refolves itfelf into the following feries 
4-&c.- 
j/i n+ 2 m ... . n 4- r+ z . m 
b 
4 * Sec, 
2 /rt . ...» 4 - >' 4 - i . m ti 4 - y n . . . . n + r 4 - 
2 . m 
~4- &c. 
71 4” 2/77 .... 72 4" 77/1 w _L. o . i _t . n . " > ~ — 
n-t $>7i _ .. . A + r+ l ..m 77 + 4//;. » + r4-2.w 
&C. &C. 
the number of feries is r, the fum of each of which being 
taken by a well known rule, the fum of the given feries. becomes 
+ . 
n. 71+771 71+r-l . 771 . 771 . r 71+771 fl + r-I .rn. 77 l.TZ 7 
c 
■ — ' - 4. 
71 4- 1771 . ... 71 + r — I . 771 . 771 . r — 2 
where the law of continuation is manifefL 
+ 
case i. To find the' fim of the Infinite feries 1 
+ 
TO 
+ 
15 
1 . 2 . 3 . 4 
+ 
^• 3 * 4*5 3 • 4 • 5 * 4.5. 6.7 
-fr&C. 
Here n 1, m~ 1, r— 3, and the third differences become 
2=0; therefore a + b 4- ic — 3, a + zb-\- 6 c = 6 , a + + i2c= 10, 
confequently = ,, b=i, c = f, and therefore the fum fought 
will be ' -* 1 ■ 1 
r • * • 3 • 3 s . 3 *. a. 2 * 3 3^ 
CASE 
