318 LAGRANGE. 



^~a''^ 0.'^' ^ 1.2 a'^' '^ 1.2.3 ^^^^ +••• 



0.^ fpo!-' t{t-Z) p'o!-' t{t-^) (t-5) for" 

 i i~^ 1.2 2'"' 1-2.3 2'"' 



If then we put in succession these values of /3* in the ex- 

 pression Ao^ ^^ we obtain two series in powers of a, namely, 



Af |a^- + ipq^^'-' + ^-^^ //a--^ + . . . J , 



and Aq-' \o^' - tjpqa''^'-' + *-^j^ fi^'^'" - . • 



Either of these series then would be a solution of the equation 

 in Finite Differences, whatever may he the values of A and a ; 

 so that we should also obtain a solution by the sum of any number 

 of such series with various values of A and a. 



Hence we infer that the general solution will be 



y., = p' {/ {^-t) + tpqfix -t-2)+ ^^t^pYfix - < - 4) 



+ ^^T4^i'W(— 6) + ...} 

 + q-' U, ix + t)- tfq ,^ (a; + « - 2) + '-^^ pY <l>{x+t-i) 



Here f [x) and <^ {x) represent functions, at present arbitrary, 

 which must be determined by aid of the known particular values 

 of Vx,^ and ?/„,,. 



Lagrange says it is easy to convince ourselves, that the con- 

 dition 2/^^=0 when t has any value greater than leads to the 

 following results : all the functions with the characteristic must 

 be zero, and those with the characteristic / must be zero for all 

 negative values of the quantity involved. [Perhaps this will not 

 appear very satisfactory ; it may be observed that q~*' will become 

 indefinitely great with t, and this suggests that the series whicli 

 multiplies q~^ should be zero.] 



Thus the value of y^^t becomes a series with a finite number 

 of terms, namely, 



