393 
the calculus of functions. 
This latter question, of far less difficulty than the former, 
was readily solved, but I did not at first observe that they 
rested on similar principles ; this, however, was pointed out by 
Mr. Herschel, to whom I had mentioned the subject. Such 
was the origin of the following enquiries. 
The question, in its most general point of view, is the solu- 
tion of functional equations of all orders. 
This, however, is a generality winch I do not pretend to 
have attained. In the first part of this Paper the reader will 
find a new method of solving all functional equations of the 
first order ; it depends on possessing their particular solutions. 
In the subsequent part, I have given various methods of solv- 
ing functional equations of the second and higher orders : 
some of these possess considerable generality ; and if we con- 
sider only those in which the n tb and inferior orders enter 
simply, such as 
F | Xy ifu?, x, . . x } = o 
I have pointed out the means of obtaining their solutions. 
The determination of functions from given conditions most 
probably took its rise from the integration of equations of par- 
tial differentials ; and we accordingly find that the authors of 
this calculus were soon engaged in the new problem to which 
it gave birth. D’Alembert was the first who occupied him- 
self with this subject : he was soon followed by Euler and 
Lagrange ; but it is to Monge that we are indebted for the 
most general view of the subject. His enquiries were directed 
to the determination of two functions from given conditions ; 
they are contained in the fifth volume of the Melanges de 
Turin , and in two Papers in the seventh volume of the Me « 
moires des Savans Etr angers, 1 773. 
MDCCCXV. 3 E 
