88 
unless the ratio of the angular momentum to the mass exceeds 
a certain limit. — Comptes Rendas , xvi., 1843; Jour, de Math., 
1851. 
In the Journal de Matliematigues for 1855 we have a farther 
contribution to this branch of hydrodynamics in the memoir 
entitled “ General Formulae relating to the question of the Stability 
of the Equilibrium of a mass of Homogeneous Liquid rotating 
uniformly about an Axis.’ 7 The memoir, “On a passage of the 
Mecanique Celeste relating to the Theory of the Figure of the 
Planets” {Jour, de Math., ii., 1837), in which he points out and 
corrects an error of Laplace, should also be mentioned. 
On dynamics we have three memoirs in vols. xi., xii., and xiv. of 
the Journal de Mathematiques, dealing with certain cases in which 
the equations of motion of a material point, or of a system of such, 
can be integrated. The equations are transformed by the substitu- 
tion of various systems of generalised coordinates (mostly elliptic 
coordinates), and then the form of the Force Function (Potential) is 
so specified that integration in finite terms shall be possible. The 
third of these memoirs, which deals with a system of material 
particles, is interesting mainly as regards the theory of Abelian 
integrals. In addition to these there are memoirs, 11 On a particular 
case of the Problem of Three Bodies,” Jour, de Math., i., 1856 ; and 
“ On Developments of a chapter in Poisson’s Mecanique ,” Jour, 
de Math., iii., 1858. 
Liouville made several contributions to Planetary Theory, among 
which which we may specially mention his memoir, “ On the Secular 
Variations of the Angles between the straight lines that form the 
Intersections of the Orbits of Jupiter, Saturn, and Uranus.” Jour, 
de Math., iv., 1839. 
In a variety of scattered notes are to be found some very important 
additions to our knowledge of Theoretical Dynamics. Perhaps the 
most striking of these is that “ On a Remarkable Expression of the 
Quantity which in the Movement of a System of Material Particles 
connected in any way is a minimum in virtue of the principle of 
Least Action.” , Jour, de Math., 1856. If we take the case of a 
single free particle, and use Cartesian coordinates, Liouville’s result 
for the form of the integral which expresses the action is — ■ 
