calculus of functions and other branches of analysis. 201 
treating them is sufficiently obvious from those already given. 
It appears then, that as in common algebra an expression may 
become illusory from the variable quantity assuming a parti- 
cular value ; so in the doctrine of functions an expression 
may become likewise illusory by the variable function assum- 
ing a particular form ; in the one case the real value may be 
a constant quantity, in the other it may be an arbitrary func- 
tion: nor ought this circumstance of the appearance of an 
arbitrary function to surprise us, when we consider that (as 
for instance in the second example) it is not one form only of 
the function fv which will satisfy the equation foe .fetoc— 1, 
but any of the infinite variety of forms comprehended in the 
expression foe = |% (a?, uoc) Y~~ ** and similarly for the va- 
lues of fee. The circumstance of this species of vanishing 
fractions having an arbitrary function contained in their true 
value, is of considerable importance in the calculus of func- 
tions, as I shall now shortly endeavour to prove. 
The Royal Society did me the honour to insert, in the last 
volume of their Transactions, a paper of mine, in which I gave 
a new method of solving all functional equations of the first 
order, and of a certain class by means of elimination, and I 
there stated that all solutions so obtained were only par- 
ticular cases of the general solutions, and that they did not 
contain any arbitrary function. 
Now it may be observed, that there are certain forms which 
may be assigned to the coefficients which render those solu- 
tions apparently infinite, yet that on farther consideration it 
appears, that the solution is in fact a vanishing fraction ; in all 
such cases the process I have just pointed out will give the 
real solution which will contain an arbitrary function, so that 
it is in fact a general solution. Thus in the equation - \ foe. 
