SERIES. 

 A B^_ C D 



X + 2X(X+ I) "^ ix(x + I) (x + 2) "^ 4.x (« + I) {x + 2) (.r + 3) "^ ^''' 

 and 



_A_ B C D 



I +.»• 2 (:l- -t- I) (4: i- 2) 3(.v+ l)(.Vi- 2) (4; + 3) 4(4;+!) . . . . (x-t-'4l + "^^ 



For by fubtraAing thefe one from the other, we have 



A B ^ C 



^•(ar+l) x{x+l}{x + 2) .v(.r+ I)(,r + 2)(jc+ 3) 



and confequently the former is the fum of that feries whofe general term is 



X {x + I) '^ X (x + I) (X + 2) x(x + I) (.V + 2) (x + 3) + 



So that the whole difficulty is now reduced to that of tranf- x^ ::= x + ^x (x — 1) + x (x — i ) ( .* — 2 ) 



forming any propofed funftion, cxprtffing the general term x'< = x + T x(x - i) + 6x (x - i) ix - 2) + x ix - i) 



of a feries into an equivalent function of one or other 01 f _ 2 W N 



the above forms. ' ^ H 



To transform a quantity of the form ^^' ^'^* 



a + bx + ex' -\- dx^ + ex^ + &c. As an example, let 



into another of the form i ^ 2 x + Ax'' 



A + B .V + C .V (.t — I ) + D A- {x - 1) (.!■ — 2) + &c. be the propofed general term. Here 



By the aftual multiplication of the latter formula, we ix — 2 x 



have , , , 



^- -^ therefore, 

 B.v = B.r 



C.x(.v- i) =-C.v + C^'' ' +3.r + 44-=i +7.v + 4.v(.v- I) 



D .V (.r — I ) ( .V — 2 ) = D .1- — 3 D .t " + D .V • which latter is of the "form required. 

 And equating the co-efficients of the like powers of x in ^^ ^'^'^'^^'^'^ ^"7 general term of the form 



this and the original feries, we obtain a + i x -\- c x'^ -\- d x' + &c. 



d=D 1 f D =d a' + b'x + c'x"- + d'x^ + &c. 



c 



\ 



S ~ l,D _ ( „r J C = f + 3 ^ i„to another of the form 



a = A J 



A ^a A , B 



+ „-r-7-7TT— : + 



Whence the values of A, B, C, D, &c. are determined by ^' ^•*' + ') " (•'" + ') ^'^ + ^) •»' (•»' + •) C^' + 2) t^' + 3) 



means of the known co-efficients a, b, c, d, &c. And j^ &c. 



the lame method may obvioufly be employed in any other 



(Imilar cafe. The following tablet, however, will facili- . ^he moll general method of performing this transforma- 



tate the operation ; -uiz. ^'"" '^' '^V '""Sual divifion to reduce it firlt to the form 



x' = x+ x{x-i) 7^ "^ "^ ■'■ T^ "*" T'" "*" *^" 



Now 

 I 



, j o 



x(x+\) _ a: (4: + I ) (.V + 2) •>; (^ + J ) (x + 2) (.V + 3) "^ .f (.V + 1 ) (.V 4. 4) 



I _ I 3 II 

 r< ~ ^^(iT'l) {x + 2) ^ x(x-^ l)[x + 2){x^l) "^ .^C*+ 1) (a-)- 4! "^ **^" 



I t 



— = ■ 4- &c &c. 



x^ xix-^ l){x-{- 2){x-t 3) 



Or by making A = a. 



B = a + ^ 



C = 2 =c + 3 ,9 4- ,. 



D " 6 a 4- 1 1 ,S 4- 67-4- i 



E= 24 a 4- 50/?+ 35) f, 0^4-1 



F = 120 a 4- 274 ,3 4- 225 y 4- 85 ^ 4. 15 , 4- e 



which 



