1817.) Royal Institute of France. 237 
the fine researches of Euler on tautochronic curves, then pass 
under review, and the author adds considerations on tautochronism 
in curves of double curvature. 
The celebrated problem of the brachystochrone, solved at first 
in an elementary manner, and then examined in a general point of 
view, leads to one of the finest methods which mathematicians 
possess, the method of variations. 
This section is terminated by the theory of the motion of a ma- 
terial point upon a given surface. The author verifies the principle 
of the least action in this kind of motion, determines the normal 
pressure of the surface, and applies this theory to the curious pro- 
blem of the pendulum with conzcal oscillations ; that is to say, which 
oscillates by turning round the vertical line drawn through the 
point of suspension. 
The third section contains the general theory of motion, both of 
a system of material points, and of a continuous solid body, attend- 
ing to their extent, and supposing their form invariable. | It begins 
with the demonstration of the general principle of motion, which 
the author applies first to several questions previously resolved, and 
in particular to the machine of Atwood. He then brings into con- 
sideration the weight of the cord and the mass of the pulley, which 
he had at first neglected ; and shows that the introduction of these 
elements would have occasioned no alteration in the consequences. 
He deduces from this principle the formulas of the motion of a 
body acted on by any forces whatever, and obliged to turn round a 
fixed axis. ‘The formulas contain the remarkable expression of the 
momentum of inertia; and the author takes advantage of the recent 
extension given to this theory by M. J. Binet. The questions re- 
specting the centre of oscillation and the compound pendulum 
naturally follow these researches. ‘After having solved these ques- 
tions, and demonstrated several curious properties in the compound 
pendulum, M. de Prony gives the theory of an apparatus which he 
contrived for giving the length of a seconds’ pendulum when men 
of science were employed in fixing the basis of the decimal system 
of measures. 
The problem of the centre of percussion, which in the 17th 
century had occasioned long discussions among philosophers, was 
solved only in particular cases. M. de Prony has supplied this 
defect by solving the problem with all the desirable generality. 
The fine and difficult theory of the motion of rotation round a 
point was studied, and discoveries made in it, by Euler, Lagrange, 
and Laplace. M. Poisson has given an excellent account of it in 
his Treatise on Mechanics. M. de Prony, taking advantage of the 
labours of so many mathematicians, and of his own at different 
times, has endeavoured to make his explanations and developements 
as complete as possible. He treats of the case in which this motion 
takes place on a fixed plane, and takes as an example the problem, 
of the top, which Kuler appears to have first resolved. 
