I 



SERIES. 



a , , , - 1 I I 



where the law of continuation is obvious, p being = - ,, and the values of ^, -^, -^, &c. being derived as follows, 



■viz. denoting thefe refpeftively by — -, — , — » &c. 



I _ I 



^ij5 = &c. &c. and generally 



« n + I (« + I ) » ^« - I) ^ («+ l) «(«- l) (/»-2) («- 3) 



--0 = — '.5 — ■ ■- ■ - ■ — - tp — ■ " .. ..., „ ui etc* 



2«+4 2 ' 2.3-4 2.3.4.5.6 



12. Fraaional Series of the folloiv'ing Form. 



m m (m + p) ^ m {m + p) (m +Jp) ^ m {m + p) . . . . (m + (n — 1) p) 



(X) = 



^ ■' y ^^(l3+p) '^ ^{^ +p)(i3 + 2p) ^ • ■ /3 (^ +/) (/3 4- (n - I) i>) 



'-./>- 



m 



(^ ) .. , -^^■- , „^ ± .. , . „^, , , , ■> + ,.^,^,^^o., ± ^'^• 



(/ + ?)('« + '•)(/+ 2 ?)(«+*'■) (/>+ 3?) ('"+ 30 - (p + qn){m + nr) 



pq 

 (30 TT^'-Tli ± ,-rT-^ + : .^' .^ 7 + Sec 



(/ + ?) /3 "^ (/- + 2?)/3' (/ + 3?)/S^ - ■ ■ ■ (/" + «?) (3" 



(S) = -L r (^"-^ "J^-'^ . (s) = _L r -^r ^ 



qJl3"(^ + «) q J J^ 



rhere /3 muft be affirmative, and not lefs than unity. 



(4-) ^7^T-T^ + n. , .^ .a , . ■^ + /-ITT-T^TTrT^rX + &c- 



. ^ a a 4- « a 4- 2 f 



^^•■' 3 (A 4- (* + 2 "^ (* + {^ + 2.)(i + 30 "^ (i + 20(^4-30(^+40 



+ &c. . . ^i^+_L)i 



Ji+ («- 0^} (^ + «0 {^+ (" + 0^} 



.„. _ {^Zah-\-ac — le')n-\- (a c + i e) ri^ 

 ^ > ~ 2b{b + c) [b + nc) (b+ (n+ .) c 



4- &c. . . 



(S) = 1 - 



a 4- ^ a + zb 



I 4-/5) ^ '(a 4. f ) (<j 4- I 4. i) (a 4- I -f 2 A) 



a 4- (« - * 



(a. 4- I) (a4-~l 4- *) • • • (a + i + {n - x)b) 

 I 



(a 4- {a + I + b) ... (n + i + {n ~ l) b) 

 (7.) o . f 4- (a -I- ^) f 7 4 (a 4- 2 *) f 9' 4- (a 4- 3 i) <: y' 4- &c. 

 • CN _ a<^ (g* — I) -I- nbcq " _ beq (q" — i) 



q-i iy-0' 



(8.) 



