4-oS Mr. vince’s new Methed of 
and are affeded with the figns -f, alternately: for 
mftance, the feries f - .* -f- l - 4 ,-f- &c. if we colled two 
terms into one, beginning at the firft term, will become 
“ ~ ' “f - &c. If we begin at the fecond term it 
2 • o 4 • 0 • 7 
becomes ~ — [-- ! 7 -f- &c. ; neither of which gives the 
ium of the affirmed feries ; but in this, and every other cafe 
ot the like nature, a correction will be neceflary : to deter- 
mine the value of which, and from whence the neceffity 
thereof arifes, is the fubjed of this third part* 
L E M M A. 
If r be any quantity whatever ^ then will — — L ~ I _p_ 1 _ L ^ 
ir r r r r 
& c. ad infinitum. 
For - = - 1 -- (by common divifion) - - - -f - - I -f&c.ad 
2 f Y “J“ y * T Y y f 
infinitum. 
Cor. 1 . Hence — — = — i -pd__ 1 y 1 - Sec. ad infinitum. 
2 r r r r r 
Cor. 2 . Hence alfo z _ * -p * _ 1 +&c. ad infinitum ; 
IV V V V V 
and - i — = - - + - - - + - — &c. ad infinitum. 
. ‘IV V V V V 
PROP. I. 
T7t 7 w 
l J — ~ — he the general term of a feries formed by writing for 
n any feries of numbers in arithmetic progrejion , , and whofe 
figns are alternately 4- and - ; then if a feries be formed by col - 
5 ledting 
