39 6 
Mr. vince’s new Method of 
y-a-b 
+ 
3. ar +• 4 / — . 1 . b 
I . / + I . 2 r+i . 3/ + I 2 r+ 1 . y+ 1 . 4 4- 1 . y 4- x 
+ &C. " 
+ 
3^*4- 8r — I . b 
4 r + 1 • 5 ' 4 - 1 . br -t-i . 7/ 4- 1 
where the numerators are in arithmetic progreffion, and the 
denominators the lame as in the given feries ; affume therefore 
ya-b = m, ya+~y- i . b = m + n, hence b = ” a = T ” l ± n - 
4 . 2 • 
I2>‘ 
lubfhtute theie values into cor. i. prop. 2. and we have 
m n 
m 
= + 
1 . > 4 - 1 . 2 4 - 1 . 3 * 4- 1 2r + 1 • 3 r 4 * i • 4 r "I" 1 • 5 r 4 " ! 
m-\-2n 
+ 
4 ^ 4- 1 . 5 r 4 - 
7^ * — +&C. ... = 4-»-3 + ? 1 » s + 
I . 6 r 4 - i . 7 ?' 4 - 1 b/ 4 
3r 4 ~ 1 • n — 2m rm + n 
6r 4 6r 3 . r 4- 1 * 
Let r = 1 , and we have 
m m + n 
' + 
1.2. 3. 4. 3.4.5. 6 + 5 .6.7.8 
I 
4 - 
12 
If m~ i,n — 1, 
m—if n~ 1 , 
m=J,n=s, 
1 1 2 c 
1 • 2 • 3 • 4 3*4*5*k + yr67778 +&c — - 3 S ~^* 
; + &c.... - * ‘ IS^ 
I 2 
■ + 
1. 2. 3. 4 1 3. 4. 5. 5+3.0. 7.0^ 
1 , 12 17 
+ ~ r + ' 
6 6 
1 • 2 • 3 • 4 ‘ 3 • 4 • 5 • 6 T 5 . 6 . 7 . 8 + &c ‘ * * * ~~ 2 S r 
Let r = 2, and we have 
«+» W2» 
1 . 
?» + » 
4 - + 
3*5*7 5* 7 *9 * 11 9* 11 * 13 . 15 
+ &c = ^Z^ x . s _i 9 »-i 6 w 
12 
288 
If m = zi , n ~ 3, 
/77— 1 , « no, 
1 -3*5*7 ‘ 5*7 *9*ii + 9^TTyT5 +&c fb~i' S 5 
41 
1 
1 • 3 • 5 • + 9 . 1 1 . rj 77 ^+^-=~ s -^ • 
m ~ ig , n ~ i 6 . 1 35 51 T 
1 • 3 • 5 • 7 5 * 7 * 9 - ii + 9*u. i2 .Tc + &c * •' = 7 s * 
Cor If * : m as 4 r : 3 r + 3> tire fum of the feries can be accu- 
rately had ; let therefore n = 4 r and m = 3 r + 2 , and vvc lhall have 
3 r +2 
