459 
Mr, Herschel on various points of Analysis^ 
C -2 
K 
} 
Now, if we make = jt, we get 
C ~ a: 4- — 1 
and consequently, 
f (*) = i { (■^ + v'-i” - 1 i + {X — i)^ }. 
If we suppose (p {x,y) to denote any function of x andy, 
and conceive this expression substituted for x, as follows ; 
<p {<f (•^> 
\vc shall liave the second partial Junction, taken with respect 
to X, which we may denote thus, ^ If we repeat, 
or conceive repeated, this operation m times, we shall have the 
mth partial function with respect to x: 
<P”’' {x,y) —<p {?>'"-*■' {x,y) ,y]. 
If the mxh partial function with respect to x be in like manner 
successively substituted ?i times fory in the expression p (a:,y), 
we shall obtain a result, 
m, n / \ f m, n— i / \ 
<p — 
and so on for more variables, z, iv, &c. — . An equation con- 
taining any number of the successive orders (x) ^ x, p [x), 
, , . , p*^ (x), of a function p, and from which p is to be deter- 
mined, is called a functional equation of the ?2th order, in p. 
Til us the equation 
0 (p'" (x) — ( 1 4- 6) . (p (x) 4" 
is a functirnai equation of the second order, and is satisfied by 
the following 
p (x) = ^ 4~ 
