* 
33% . FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
FOURTH SERIES. 
Figures of equilibrium of revolution, other than the sphere and the cylinder. 
§ 1. The preceding series having completed the theoretical study of the liquid 
vein, we return to liquid masses withdrawn from the action of gravity, and pro- 
ose to prosecute the examination of figures of equilibrium of revolution. 
Let it be remembered, in the first place, that if we designate by R and R’ 
the two principal radii of curvature at the same point of the free surface of a 
liquid mass virtually without weight, and by C a constant, the expression of 
the general condition which such a surface should satisfy in a state of equili- 
briuam is (2d series, § 5) atpae: an expression in which R and R’ are 
positive when they pertain to convex curvatures, or, in other words, when they 
are directed to the interior of the mass, and negative in the opposite case; let 
it be also remembered that this equation is a simple transformation of that 
which implies that the pressure exerted by the liquid on itself, in virtue of the 
mutual attraction of its molecules, does not change from one point to another 
of the surface of the mass, (2bid. ;) and be it remembered, lastly, that, accord- 
ing to a known property of surfaces of revolution, if the figure of equilibrium 
pertains to that class, one of the radii R and RY is the radius of curvature of 
the meridian line at the point under consideration, and the other is the portion 
of the perpendicular comprised between the point in question and the axis of 
revolution, or, as may be expressed more simply, the perpendicular to that - 
ojnt. 
é In this case, that is, in the case of surfaces of revolution, the preceding ex- 
pression, put in the differential form, is completely integrable by elliptical func- 
tions, so that the forms of the meridian lines may be deduced from it, and it is 
this which M. Beer has proposed to do in a recent memoir,* in which, for the 
second time, he has done me the honor of applying the calculus to the results 
of my experiments; and, besides this, a property discovered by M. Delaunayt 
by means of the calculus, and since demonstrated geometrically by M. La- 
marle,t enables us to attain the same object without having recourse to ellipti- 
cal functions. We shall speak, in a proper place, of these resources of analysis 
and geometry; but, in the present series, we purpose to arrive at the forms of 
the meridian lines, at all their modifications and all their details, by a reliance 
upon experiment and by availing ourselves of simple reasoning applied to the 
relation which the equation of equilibrium establishes between the radius of 
curvature and the perpendicular. Our undertaking, in which experiment and 
theory will proceed side by side, may thus serve as a verification of the latter. 
To avoid all ambiguity, we will replace the letters R and R’ by the letters 
M and N, the first of which will be understood to designate that one of the 
two principal radii of curvature which pertains to the meridian line, and the 
second that which constitutes the normal or perpendicular; so that, as regards 
figures of revolution, the general equation of equilibrium will be, aty=e- 
§ 2. This notation being adopted, we shall proceed, first, to demonstrate that 
the sphere is the only figure of equilibrium of revolution whose meridian line 
meets the axis. ‘To this we may add the plane, if we consider it as the limit 
of spheres, or as the surface generated by a right line perpendicular to the axis. 
*Tractatus de Theoria Mathematica Phenomenorum in Liquidis Actiont Gravitatis De- 
tractis Observatorum. Bonn, 1857. 
tSur la Surface de Revolution dont la Courbure Moyenne est Constante. Journal de M. 
Liouville, 1841, t. vi, p. 309. : 
+ Theorie Geometrique des Rayons et Centres de Courbure. Bulletin de l’Acad., 1857, 2d 
series. 
