4 03 
Mr . vince’s new Method of 
mzz i, « — i 
.. 
i • 2 . 3. 4 -5 
j_ — 4 
. 3 - 4 * 5 * 6* 7 5* 6.7- 8.9 
1 
1 1 ^ 
1. 2. 3.4-5 
‘ 3 - 4 - 5 - 6-7 ‘ 5. 6. 7. 8. 9 
1 
_j 
, - . + ——4 
25 
41 
4 -&-C. . . . rr -S — - ; 
3 9 
+ &c. . . , = | d S ; 
•&c» . . . ~ - S — . 
3 3 ^> 
57 
CT = 25 ,»=: 16,- ' " g +~ A C ft - + — 5 +&C....-=-S. 
1 • 2 • 3 . 4 • 5 ^ . 4 . 5 . o . 7 5 • ® • 7 • & • 9 3 
Let r— 2, and we have 
m 
7 . b . 9 
1 • 3 • 5 • 7 • 9 5.7.911.13 
~ -p & C ' • » r — 
9 . 11 . 13 . 15 . 17 
If/«= 1 , o, — — 4 
4 .w— 47 
iy 2 
+ 
x S + 
5 q*- 44 w 
24 . 120 
* - 3 - 5 - 7-9 9 • 1 1 • i 3 1 5 . 1 7 
1 
x 1 1 
+ 5 cc....r= -~S- 
'48 24. 30* 
I I 
— — 2,- — ; ; — “ 4 
: i 7 
4&c.... — — s- ' 
1 • 5 • 7 • 9 9 • * 3 • 1 5 * 1 7 96’ 24.60' 
Cor. If » : as ir : zr + i , the fum of the feries can be accu- 
rately funnel ^ ti fl u n 1 6 th.erctore n — 2r, w ~ 2r + i , and we have 
i 
-& C. . .. 
J.r+i .3^4-1. 4r+x 2 r -f- 1 . 3 r + I • 5 r + I • 6r 4- i 6 .r.r-j- I . I 
Ii: ' ' “ 1 ’ 1.2.4 -5 + 3~. 4 . 6 . 7 + 5.6. 8 .9 + &c “ Jb * 
— 1 1 1 1 
T ~ 3 ' 1.4. 10. i3 + 7*.io‘. 16 . 19“^ 13 1 16.22.25 + <ScC * 
Having thus far explained the method of fummation 
of fuch feries as I propofed to treat of iti the firft part of 
this paper, I trufi it is not necefTary to fay any thing fur- 
ther, as the fame method of proceeding will manifeftly con- 
tinue the feries to any propofed number of-fa&ors in the deno- 
minator; I fhall therefore conclude with pointing out a re- 
mai table piopeity of thofe feries whole fum can be accurately 
found . that when the number of factors in the denominator 
is even, the numerator is always equal to the fum of the two 
5 middle 
