606 Proceedings of the Royal Irish Academy, 
6. Let us now consider D''f{x**), We shall show that this can be 
reduced to the question of finding B'^e'"''*. We first give one or two 
further theorems which will be employed. 
If <^,/are any two functions 
For either may be expressed in the form 
£1 
e^'dy ^ {^x)f{y) = <p {x)fy + <p'{x)fiy) + i <p"{x)f"{y) + . . . 
Hence if / be a function such that it and its coefficients do not become 
infinite for y = 0, 
Also, if both /, ^ be such functions, 
See Boole's Finite Differences, by Moulton, p. 23. 
We may express the function f{x + <l>{x)) in the form 
e^^^)^f{x)=f{x + <!>{x)), (12) 
the operation indicated being performed as if x were a constant inde- 
pendent of X, which is replaced by x after the operation is concluded. 
If we put 
d 1 d 
fj = x^, — = ; 
dy nx'*-'^ dx 
hence, putting ^ for the operator, 
^=^-i^.„+ijr>^ (13) 
e dyf[y)=eHf{x;'^)=f{x^-^h), 
e'^^^^^f{^)=y{x-+c!>{x)). (14) 
Also " =/i(^"), =/2(^**), &c. 
To find now the value of D^fix""), we have (10) 
