460 Mr. Herschel on various pohits of Analysis, 
An equation between -any number of the partial functions 
^m,n, 8 cc (x, y, &c) for determining the form of (p (a:,y, 
&c) is called an equation of partial functions, and its order 
may be denoted in the same manner. Thus 
0 = ' {x,y) + (f.’’' {x,y) _ (a + 6 + 1) . (f, [x,y) +c 
is an equation of partial functions of the second order wdth 
tw^o variables, and is satisfied by the equation 
<P =ax + by + c 
Let (p (x) be a function of x, and a certain number of con- 
stants dybjC,, . . And from this expression conceive [x), (p^ 
(x) &c to be successively formed which will be functions 
also of a, b, &c. If the number of these constants be ?i, we 
may thus produce n 1 such functions of them, w'hich will 
be respectively equal to the several orders of (p which they 
represent. Thus we have + 1 equations involving the n 
quantities a, b, c, %i<z which may therefore be eliminated, and 
the resulting equation between x, (?>“ (x), (p^ (x) &c will 
therefore be independent on them. As far then as regards 
this equation they are arbitrary, so that in reascending from a 
functional equation which contains n 1 different orders of (p 
(not including <p^ (x) or x) n arbitrary constants must be 
introduced. The reasoning here made use of is sufficiently 
plausible, and in fact, no other than has been adduced in de- 
monstration of well known and important truths. The con- 
clusion too, to the extent of its literal meaning, is correct. 
But we have here to notice a paradox of a very singular nature, 
\iz: that even in the simplest cases imaginable (such as (x) 
= x) the general expression for (p (x) may contain, not one 
or two, but an unlimited numler of arbitrary constants, nay 
