EQUATIONS IN FINITE DIFFERENCES. 95 



To determine Z,, Z^, &c. put <p{A) = e'"*, a being an arbitrary quan- 

 tity. 



Hence by substitution we have 



Now the value of the left-hand member of this equation is also 



And if we equate the coefficients of 2" in each we get 



2 _^ S^CD' „ 



1 .2...6 X I.2...CX 1.2...(/x...'^ 



subject to the two conditions 



ia + b + c + d + &c. = n 

 \ b +2c+3d + kc. = m. 



These conditions being satisfied we obtain the identity 



1 .2...« X 1 . 2...C X 1 .2...a T- V / 

 and comparing this result with the first expansion, viz. 



A + Byz + Cy'z' + By^z' + + My-Z'" + &C. 



BCD'' 



we have M = 2 — - — , , ^T '"' — j — s— x 0'"*"^ ; 



1 .2...i X 1 .2...CX 1.2...rfx &c. ^ ' 



where it must be observed, that b, c, d, &c. having previously satisfied 

 the equation i + 2c + 3t?, &c. = m, the quantity n is then found by 

 summing b, c, d, &c. 



Put C = c,B\ D = cB', E = c,B', &c. 



and making m = 1, 2, 3, &c. successively, we get the following identities, 

 by which A, y, c,, d, C3, &c. are completely known. 



