394 
Mr. Babbage’s essay towards 
In the first of these he explains the solution of several 
functional equations by means of curves of double curvature, 
and by curve surfaces. 
In the second Paper, the question is treated in a more ana- 
lytical method, and he endeavours to reduce it to the solution 
of equations of differences. “ Je me propose,” observes Monge, 
“ de faire voir que la determination des fonctions arbitraires 
qui se trouvent dans 1’ integrale d’une equation aux differences 
partielles, depend en general, dans les cas que je n’ai pas 
encore trails, de 1’integrale d’une ou de plusieurs equations 
aux differences finies, dans lesquelles le rapport de la vari- 
able principale a sa difference finie est donne soit qu’il soit 
variable soit qu’il soit constant.” 
In the same volume is a paper of Laplace on this subject, 
which he views in the same light, and endeavours to reduce 
functional equations of the first order to those of finite diffe- 
rences. This skilful analyst first solved the functional equa- 
tion F (px, (pa x, } = o. The method he made use of is 
peculiarly elegant ; he converted it into an equation of finite 
differences in which the difference was constant. Still, how- 
ever, it appeared by no means the most direct method to make 
use of such an expedient, nor was it even known that all equa- 
tions of the first order admitted of its application. This latter 
objection was, however, removed by Mr. Herschel, who in 
an excellent paper on functional equations, has extended the 
method made use of by Laplace to the solution of all equa- 
tions of the first order. His solution is equally elegant and 
general; it leaves nothing to be regretted, but the narrow 
limits of our knowledge respecting the integration of equations 
of finite differences. From this and other causes, I am still 
