236 Proceedings of Philosophical Societies. {[Marcn, 
the reciprocal actions of bodies on each other, the theorem of the 
equilibrium of two material points which strike each other in oppo- 
site directions, “and the formulas of this motion when equilibrium 
does not take place. These formulas are applicable to any degree 
whatever of elasticity, and he deduces from them with facility and 
as corollaries the cases of perfect hardness and perfect elasticity. 
He demonstrates occasionally some theorems respecting the preser- 
vation of the motion of the centre of gravity, the preservation of 
living forces, the loss of force in the case of sudden change, the 
principle of the least action. Among the examples of the theory 
of compound motion are found the determination of the angle of 
reflexions for any degree whatever of the elasticity of the body re- 
flected, the examination of a case of motion which renders the 
difference between impingement and pressure sensible, and the 
properties of the machine of Atwood. He shows how the experi- 
ments should be made, and gives the requisite formulas for calcu- 
lating them. 
The second section treats of the motion of a material point, 
attending to the different conditions to which this motion may be 
subjected; those, for example, of moving either freely or along a 
given line or surface, or so as to satisfy certain conditions. Among 
the details which may be useful to those destined to cultivate the 
sciences will be distinguished the verification of the principle of 
areas, the relations Letween the areas and the momenta. 
The author passes to the motion of heavy projectiles ina vacuum. 
He demonstrates analytically all the theorems of Galileo; and 
among the particular problems on that subject there occurs one 
whose solution is connected with the particular solutions of diffe- 
rential equations. 
The fundamental equation for this motion does not suppose at 
first any hypothesis respecting the law of resistance. The author 
then introduces into it the law of the velocity, and deduces from it 
all the formulas necessary for those who have the management of 
artillery, treats of the resilience employing the simplifications 
allowed by the smallness of the angle of projection, and Jays down 
the principal bases of the calculus to obtain the absolute numerical 
results for trajectories described in resisting mediums. 
To this useful explanation one of great interest succeeds, that of 
the motions of the planets whether elliptical, parabolic or hyper- 
bolic. This part of the work is so arranged as to constitute an 
elementary introduction to physical astronomy. f 
The motion of a heavy moveable body on a polygon serves as an 
intro.luction to the explanation of the general properties of a body 
moving in any curve whatever, fixed and continued. From this is 
deduced the theory of the simple pendulum, and the principal 
philosophical truths of which this instrament has occasioned the 
discovery or confirmation. 
The eycloid and its properties, the discoveries of Huyghens, and 
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