> 9 ^ 
Mr. vince's new Method of 
Cor. i. la prop. 3. write a for wand 2 b for n, and we have 
— * + - a + 2b - _> & c _ 
i.r+i.2r-j-i.^r+i 2r + i . $r-t- i . 47-4- I . $r + I 
~ ra ~ 3 r + 2 . b ^ gr-f 1 ■ b — r a ra 4- 2 b 
3 r+ X 3 r * ~ oTTiTI * 
Alfo in this prop, write a + b for m , and zb for n, and we have 
a-\- b ' 
a + $b 
— - ■ ’ . ■■ ■■ ■ i — - -}- --- , . _J_ &CC — 
r + 1 . 2 r-\- i • 3 r + 1 • 4 r 4“ I 3r + I . 4r + I . jr + i . 6r-{- i 
3' + 2 • h ~ 2rg x g _i_ i . b ra + 2 b 2ra + b 
3 } 6r 3 . r-(- l 6r 2 . r-j- I . 2r-f- i 
iubtraft this latter fenes from the former, and we have 
=r~J=r-= - i±j + SCC = 
1 * r + 1 • 2r + i . 37- 4-1 7-4- l . 2r + i . y + I . 47- 4- i 
4 r <7— 3 r 4~ 2 . 2 b c ^r-f i . 2 b — 2 ra _ ra-\- 2 b 2 ra + b 
3 r * 37-* r 
Let r— 1, and we have 
a 
a-\-b n + 2 b 
4 - : — &c. . . . 
1.2. 3 . 4 2 . 3 . 4. 5 3 . 4. 5. 6 
I I 
3 r 3 .r+i 6r 2 .r + i . 2r+ i * 
4 g - *o l x g + ^ 3 ^~ 32 ^ 
Tf a— 1, b~ O, 
0=3, 3 =i, 
■&C. ... — hs - 
1 • 2 • 3 • 4 2T3T775 T 3 . 4 . 5 . 6 
1 &c 13 
1 * 2 * 4 a- 3-5 3 - 4-6 * C 3 ^6* 
Let r = 3, and we have 
I - a + 2b __*2a—22b 1027^-1 c6a 
1 • 4 . 7 . i° 4 .7.io..3 + 7 .i 0 . I3 .,6 + J757T^T 
= 4 + 1 - 8 7 » »o 
1 . 4 . 7.10 4.7. IO. I3 7.10.13.16’ 
* 124,3 = 3, 
'3-8 i . 5 6 3 . Si 
— 1 l &c _ 1 3 2 c 
1.7. 10 4.10.13 7.13.16 &C 72 27 S * 
Cor. 2. If a : b as 3 r+. 2 : 2 r, the fum of the feries can be ac- 
curately found ; take = y + 2 , and b = ar , and we i hall have 
3 ^ +2 47-4-2 
1 .r+i ,2r+i >3r+i r+i . ar + 1 . 3 r+ 1 .4^7 
+&c. . . . 
r+ 1 . 2r-f 1 
If 
