invefii gating the Sums of infinite Series . 
If the latter ferles had been added to the former a feries 
'would have been formed whofe terms would have been all po- 
fitive ; but as I purpofe, in the fecond part of this paper, to 
give a general method of fumming all l'uch feries, I fhall not 
ilop here to apply this method of invefligation. 
Cor. 2. In proportion 2. fubflitute a for m, and 2 b for 
n, and we fliall have 
a a-\-2b 
r-f- 1 . 2r + 1 . 3/ + 1 y + I . 4' + I .5^4-1 
+ &C . . . . — 
2 > + I . b — ra q ra — y+ 1 . b ra — r—\.b 
X O 4 T! r 
2r 2 
2 1 ~ . r-p I 
Alfo in prop. i. write a - h fo r m, and ib for «, and there refults 
a b 
+ 
a + 3 h 
2r+ l . y+ l . 4-r+ 1 4r+ I . y-\- I . 6 r + I 
a — b 
-P&C. . . . = 
ra — 2 r + 1 . b 0 . r + I . b — ra 
5 xb-| , - . 
2r 3 r-pi . 2r+ ! 
Subtradl this latter feries from the former, and we lhall have 
a a + b a + 2b 
oCC# • • • 
r+I . 2r-fl . 3r+l 2 r + 1 . y -f 1 . I 3;-+ 1 . 47-+ I . y + 1 
a — b 
2r+ i . 2 b— 2 ra _ c , ra — 37-4- r ,b f ra — r — \ . b 
— X O + _ x “ T x ; -f* 
2 r . r 4- 1 
r+ I . 2 r 4- I 
PROP. III. 
To find the fum of the infinite feries 
m 
I . r 4- 1 . 2 r 4- 1 . y 4- 1 
+ 
m+ n 
m+ 2 n 
2 .r 4- 1 . y + 1 • V + 1 • 5 r + 1 4 r + 1 • 5 r + 1 • 6r 4- 1 . 7 r + 1 
This feries refolves itfelf into 
c a-\-b , a 4 - 2 b 
+ &C. 
+ 
i.r+i. 2 r+l r+l. 2 r+l.y+l 2r + I . y -f 1 . 4r + I 
for by reduction it becomes 
F f f 2 
-&c. 
1 
ya-h 
