458 Mr. Hekschel on various points of Analysis, 
III. On Functional Equations, 
The determination of functions from given conditions is a 
point of such importance, not only in the partial diiierential 
calculus, but also in a variety of other branches, that it has 
occupied the attention of the most eminent Analysts, and it 
must be confessed, not without considerable success. Their 
researches, however, have hitherto extended no farther than 
to such conditions as involve only the unknown function, (p 
without any of its superior or inferior orders, p^, . , , &c, 
p~^\ &c. It is to equations of this latter kind, therefore, that 
we now propose to direct our attention. 
The successive orders of any functiony' (.r) may be pro- 
duced, either by actually writing/ (x) for a: in the expression 
of/(x), in which case the general value of (x) must be 
concluded from induction; or more elegantly by the following 
method. 
Assume/* (.r) = and we have/*"^^ (x) = 
0 = u , — f [u ) 
an equation of differences whose integral will be of the form 
C being an arbitrary quantity independent on %. Let % :=. 0, 
and we have 
F (0, C) = =J^ (x) = X 
an equation which gives C in functions of x. 
For example ; let J (x) = 2x* — i , and we have 
0 ■=. u , — 9 . 11 ^ 4 - 1 
and integrating. 
