266 Analysis of the Mccaniqae Celeste of M. La Place. 



which reduce it to this, namely, The sum of the products of 

 each force by the element of its direction is null. He teaches 

 ns to determine, when the point is not free, the pressure ex- 

 orcised by it upon the surface or upon the curve to which 

 it is subjected. Considering afterwards the point in the state 

 of motion, he seeks the relation which exists between the 

 forces that animate it and the velocities which should re- 

 sult from it; and byavery delicate analysis, and considera- 

 tions drawn from experience, he demonstrates that, in na- 

 ture, this relation of the force to the velocity is the propor- 

 tionality. After having developed the immediate conse- 

 quences of this law, the author gives the equation of the 

 movement of a point animated by any given forces, and de- 

 termines the pressures exercised by this point upon the sur- 

 face or upon the curve to which it may be subjected. He 

 afterwards makes the application of these principles to the 

 motion of bodies animated by gravity in a resisting medium, 

 and to that of a point gravitating upon a spherical surface. 

 The isochronism of the very small oscillations of this move- 

 able point leads to the problem of tautochrones which the au- 

 thor resolves, in the case where the resistance of the medium 

 is proportional to the two first powers of the velocity. He 

 is afterwards occupied with the conditions of the equilibrium 

 of any system of bodies considered as points : he writes 

 down for each of them the equation of the equilibrium ; and 

 uniting these results, he extracts from it the principle of the 

 virtual velocities, which is thus demonstrated in a direct and 

 general manner. After having shown how we deduce from 

 this the reciprocal actions of the bodies of the system, and 

 the pressures which they exercise upon external obstacles, 

 he makes the application of them to the case in which all 

 the points of the system are invariably united together; and 

 this leads him to treat of the centre of gravity. The author 

 afterwards considers the conditions of the equilibrium of 

 fluids : the property which characterizes them being a per- 

 fect mobility, it is necessary, in order that a fluid mass be 

 10 equilibrium, that each of the molecules composing it be 

 in equilibrium in virtue of the forces which animate it. The 

 author, setting out from this principle, determines the rela- 

 tion 



