1816.] Royal Institute of France. 151 
known methods could not give; for though we should answer nega- 
tively to these questions, the author will still retain the merit : 
1. Of having constructed by an uniform method a set of general 
formulas proper to transform definite integrals, and to facilitate 
their determination : 
2. Of having first remarked that a double integral, taken within 
given lines for each variable quantity, does not always give the 
same result in the two ways of effectuating the integration : 
8. Of having determined the cause of this difference, and of 
having given its exact measure by means of singular integrals, the 
idea of which belongs to the author, and which may be regarded as 
a discovery in analysis : 
4. Of having given by his methods very remarkable new formulas 
of integrals, which may indeed be deduced from known methods, 
but which nobody had hitherto obtained. 
It appears to us, on all these accounts, that M. Cauchy has 
given in his researches on definite integrals a new proof of the 
sagacity which he has shown in several other of his productions. We 
think, then, that his memoir is worthy of the approbation of the 
Class, and to be printed in the collection of Savans Etrangers. 
Memoirs of M. Jacques Binet, which treat of the Analytical 
Expression of Elasticity, and of the Stiffness of Curves of Double 
Curvature. Commissioners, MM. Carnot and Prony, reporter. 
After an introduction purely geometric, containing formulas 
partly new, relative to polygons whose sides are not all in the same 
plane, and to curves of double curvature, the author, passing to 
problems of equilibrium, which constitute the particular object of 
his paper, successively introduces the consideration of the action of 
forces upon a polygon of the kind of which we have just spoken, 
and upon a curve of double curvature, establishing in both systems 
a theory applicable both to the case of stiffness and to that of elas- 
ticity. 
The nature of these researches renders it impossible for us to 
analyse the memoir. We shall be even obliged to abridge the ob- 
servations of the Commissioners. We shall say only, with them, 
that M. Binet has the merit of having introduced explicitly and 
completely into his analysis all the elements of the question of 
which he treated. We say explicitly, to distinguish the method 
which he has followed from that pointed out by Lagrange’s beautiful 
method of indeterminate quantities. M. Binet combines from the 
first with the external forces those which he calls internal, which 
represent the effects of the three elasticities of an elastic curve, or 
the different efforts which tend to change the form of a stiff curve, 
In this way we know beforehand, and never lose sight of, the signi- 
fication of the signs which represent each quantity. The function 
which fills that quantity in the system is always known without 
equivocation. ‘ 
However, by this method of proceeding it is necessary to give an 
account beforehand of all the phenomena to which the combined 
5 
