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III. Experimental and Theoretical Researches on the Equili- 

 brium-Figures of a Liquid Mass without Weight. — Fourth 

 Series. By M. J. Plateau*. 



On the equilibrium-figures of revolution which are neither spherical 

 nor cylindrical. 



HAVING completed the theoretical investigation of a jet of 

 liquid in the preceding series fj I return, in the present 

 series, to liquid masses without weight, and continue the exa- 

 mination of equilibrium-figures of revolution. 



It will, in the first place, be remembered that the general con- 

 dition satisfied by the free surface of a weightless mass of Uquid 

 in a state of equilibrium is expressed by 



where R and R' are the two principal radii of curvature at any 

 point of the surface, and are considered positive or negative 

 according as they are directed towards the interior or exterior of 

 the mass. C I'epresents a constant. 



It is well known that, in surfaces of revolution, one of the 

 radii R, R' is the radius of curvature of the meridian curve at 

 the point under consideration, and the other is the portion of the 

 normal intercepted between this point and the axis of revolution, 

 or, according to a simpler mode of expression, it is the normal 

 at this point. On this account, and in order to avoid all ambi- 

 guity, I replace the letters R and R' by ]\I and N, whereby we 

 shall be reminded that the former represents the radius of cur- 

 vature which belongs to the meridian curve, and the latter the 

 one which constitutes the normal. In this notation, therefore, 

 the equation of equilibrium is 



Mathematicians are aware that the quantities M and N can 

 be expressed by differentials ; thus transformed, the above equa- 

 tion is completely integrable by elliptic functions, so that the 

 nature of the corresponding meridian curves can be deduced. 

 This forms the subject of a recent memoir by M. Beer, wherein, 

 for the second time, the author honoui-s me by applying the cal- 

 culus to the results of my experiments. The same object, too, 

 may be attained without the aid of elliptic functions by means 



* The ori<;inal memoir will be found in the thirty-first volume of the 

 Memoires de I'Acade'mie de Bruxelles. The abstraet, of which a translation is 

 here ^iven, appeared in tlie Annules de Chimie et Physique, May 1858. 



t Sec Phil. Mag. S. 4. vol. xiv. p. 1. 



