calculus of functions and other branches of analysis. 211 
perhaps be desirable to confine the meaning of particular solu- 
tion to the definition which has just been given. It is need- 
less to add that the different species of solutions just enurne- 
rated, bear a strong resemblance to those of differential 
equations. 
Amongst differential equations containing more than two 
independent variables, a large proportion do not admit of any 
integrals, these can only be integrated by assigning some 
relation between the variables, an analogous case occurs with 
respect to functional equations, a large number of those which 
contain two or more variables admit of no solution, unless 
we assign some relation between the variable quantities; 
examples of such equations may be found in the second part 
of the Essay towards the Calculus of Functions. And here 
perhaps it may not be misplaced to state a difficulty of a 
peculiar nature with respect to functional equations which are 
impossible ; for the sake of perspicuity I shall consider a very 
simple case 
for x substitute ~ ^~==c^/ x 
and by multiplication $x x ^7 — c * ~ * 4^ or 1 = from 
which it follows that c = + 1 or in other words that the 
equation $>x — c-ty — is contradictory unless c = +i. Now 
the functional equation F j x, -tyx, iS/ax j = 0 has been re- 
duced by Laplace by means of a very elegant artifice to 
an equation of finite differences, nor am I aware that this 
profound analyst has pointed out any restriction or any 
