39 * Mr. Vince’s new Method of 
Cor. i . In prop, i . fubftitute a for m 9 and 2 b for n 9 and- vve have 
a-\-lb . a-\-\b 
~ ' ~ ~{~ - : - 4 - ' 
l.r +t 2 ^ + 1 2r+ I . 3 r+ I . 4' + I V + I . 5; * 1 . 6/ + x 
+ 
; 4* 8cc, , 
ra — r+i . b ^ g _|^ 2r + I .b — ra 
2 t : 
Alfom this prop, fubflitute a + b for m 9 and 2b for and we haw 
a + i ‘ + ih +&C = 
+ 
r -f I . 21 4- 1 . 3, 4. 1 3 r + 1 . 47 + 1 . 5, 4 i 
ra+b 
r + 1 . b — r a q ra — 2r + 1 . b 
x o + — 5 p 
2 r 3 
2 r 1 x/ + I 
Subtiadf this latter feries from- the former, and 
2 ' + * + ^ &c....= 
J.r+i.^ + i r-f 1 . 2 r+l . y + 1 2 r+i .37-+1 . 4 r+i 
ra-\-nh 
2 r 2 Xr-\ 
2 ’-a — >-fi . 2 b ^ 2^+1 .b — ra ra-\-2h 
r S r**» \ “ 
arxr+i 
Let r — 1 , and we have 
. Sc c. • • . — zci — fb x S -f 
a + b ^ a-\-2b 
nb-$a 
-&c =il-6S. 
4 
I.2.3 2. 3 .4 3.4.5 
If a = 1 , b~ q, — 1 — 1 1 1 c~ __c 5 
1,2.3 2 -3*4 + 3-4-5 & - -2S- — ; 
tf=I, £=2, — - , 
i - 2-3 2.3.4 3.4.5 
Let r = 3, and we have 
1.4.7 4 - 7 • 10 + 7 • 10. 13 — xS+~-^-l_. 
If ^=1,^-0, — -| 1 g— _ 2 p 11 
1 - 4-7 4 * 7 • 10 7 . 10. 13 OCC. * . . — - o - — ; 
a - 1 9 b-\ 5 + — 3 & c __ 5 2 Q 
1.4-7 4.7.10 7.10.13 at ~~ 54 "" 27 * 
If, infteadof fubftituting in prop. i. a „d « f or „ and m . 
we had fubftituted two other quantities, as 2 r and s, and then 
proceeded, as above, a feries would have been formed, the nu- 
merators of whofe alternate terms would have formed each a 
a feparate arithmetic progreffion. 
2: 
If 
