152 Proceedings of Philosophical Societies. [Fes. 
play of forces and resistances may give place, which is not always 
easy. While the method of indeterminate quantities, which re- 
quires only the knowledge of the conditions to which the composi- 
tion of the system is subjected, is what may be called a general 
instrument, and of a usage always certain, which conducts: the 
analyst by the simple mechanism of the calculus to the discovery of 
those quantities representative of the different effects supported by 
the system. But if the method of Lagrange dispenses with regard 
to these indeterminate quantities with a preliminary exercise of the 
judgment, on the other hand it requires considerations often delicate 
in their explanation when we have equations containing them. 
From this it may happen sometimes that the explanation of which 
we speak is neglected or omitted, which renders the solution in+ 
complete. The problem of the equilibrium of the curve, whether 
rigid or elastic, furnishes us with an example of this. Lagrange 
arrives at three equations absolutely identic with those of M. Binet. 
These equations contain three indeterminate quantities, which 
ought to be referred to the extensibility, flexibility, and tortion. 
But this explanation is not found in the Mecanique Analytique, and 
it must have in general escaped those who have perused that book. 
The reporter is even of opinion that Lagrange has not paid atten- 
tion to the part which these three indeterminate quantities play in 
his system, considered under a mechanical point of view; for he 
has not said that the last two would furnish infinite forces, the one 
of the first, the other of the second, order; a circumstance which 
must have appeared singular to him if he had not overlooked it. 
M. Binet deduces these values very simply from his analysis, and 
solves very clearly the kind of paradox which they present; so as to 
leave nothing to desire respecting the value, the signification, and 
the functions of these quantities. If to these considerations we add 
the remarkable indetermination of the internal forces, which he 
first, as far as we know, pointed out in treating of the equilibriuna 
of the polygon, we shall conclude from them that he was in the 
right to announce that his researches might serve as an explanation 
and supplement of several chapters in the Mecanique Analytique. 
In general the analysis is managed with much skill; and the 
geometrical introduction, which would itself be an interesting 
memoir, ought to confirm, and even increase, the good opinion 
which has been formed of his scientific merit, from the different 
works which he has before submitted to the judgment of the Class. 
In consequence of this report, the Class, in praising the memoir, 
ordered it to be printed in the collection of Savans Etrangers. 
Memoir relative to the Stability of Floating Bodies, by M. 
Charles Dupin, Captain of the First Corps of Maritime Engineers, 
and Chevalier of the Legion of Honour. MM. Sané, Poinsot, 
and Carnot, reporter. 
This subject, treated by Euler and Bouguer, appeared exhausted. 
M. Dupin, by employing a method not known in the time of these 
two illustrious mathematicians, has obtained new results. The 
