150° Proceedings of Philosophical Societies. [Fer. 
Memoirs relative to the Integration of Equations with partial 
Differences, by M. Ampere. Commissioners, MM. Legendre, 
Arago, and Poisson, reporter. 
The author explains, in the first place, the general considerations 
which belong to equations of all orders, and which he afterwards 
particularly applies to those of the first and second order. On this 
subject, the difficulty and importance of which are known to mathe- 
maticians, such considerations are not without utility, and may 
serve to elucidate some points of theory, even when they do not 
lead to a new method of integration. This memoir, of which it is 
impossible here to give the analysis, contains, says the reporter, 
new and interesting views respecting the calculus of partial diffe- 
rences ; and the conclusion of the report is, that the author should 
be induced to continue these researches, and to connect them to 
some one of the applications of analysis to mechanical philosophy. 
Memoir on the Integration of Equations with partial Differences, 
by M. Ampere. Commissioners, MM. Arago and Poisson, re- 
porter. 
The author considers a class of equations with partial differences 
of the second order with three variable quantities; namely, linear 
equations with respect to their greatest differences. The most 
general of this class contains four terms, three of which are multi- 
plied by second differences, and the fourth is independent of them. 
The coefficients of these four terms are any functions of the three 
variable quantities, and of the first two differences. M. Ampere 
proposes to transform this equation into another, which contains 
only a single second difference, and he succeeds in his attempt. 
For that purpose it is necessary, in the case considered by M.. Am- 
pere, to change at once the three variable quantities; and the 
choice of the unknown quantity which must be taken for the prin- 
cipal new variable quantity, constitutes the difliculty of the problem. 
He gives in his memoir different examples of equations which be- 
come entirely linear by means of his transformation, and conse- 
quently integral, by the methods known. Thus, concludes the re- 
porter, though this transformation be not always practicable, it 
gives, however, a real extension to the means of integration known 
at present. It may be often useful, and contribute to the progress 
of this part of the science. In consequence, the memoir has been 
thought worthy of entering into the collection of those which have 
been approved by the class. 
Memoir on Definite Integrals, by M. Cauchy, Engineer des 
Ponts et Chaussées. Commissioners, MM. Lacroix and Legendre, 
reporter. 
As it is impossible to give an analysis of this work here, we shall 
satisfy ourselves with stating the conclusions, which are sufficiently 
extensive to save usa formal extract. 
We shall not examine if the new methods of M. Cauchy are 
more simple than those previously known; if their application: is 
easier, and if they are capable of leading to results which the 
