'Royal Society, 455 



of the fluid, and of all the level surfaces underneath it, are derived, 

 in that theory, from the single expression of the hydrostatic pressure, 

 and are entirely dependent on the differential equation of the surface. 

 Thev require, therefore, that this latter equation be determinate and 

 explicitly given ; and accordingly they are sufficient to solve the 

 problem when the forces are known algebraical expressions of the 

 co-ordinates of the point of action ; but they are not sufficient when 

 the forces are not explicitly given, but depend, as they do in the 

 case of a homogeneous planet, on the assumed figure of the fluid. In 

 this latter case, the solution of the problem requires, further, that the 

 equations be brought to a determinate form by eliminating all that 

 varies with the unknown figure of the fluid ; and the means of doing 

 this are not provided for in the theory of Clairaut, which tacitly as- 

 sumes that the forces urging the interior particles are derived from 

 the forces at the upper surface, merely by changing the co-ordinates 

 at the point of action. In the case of a homogeneous planet, the 

 forces acting on the interior particles are not deducible, in the manner 

 supposed, from the forces at the surface. 



After showing that the equilibrium of a fluid, entirely at liberty, 

 will not be disturbed by a pressure of the same intensity applied to 

 all the parts of the exterior surface, the author considers the action 

 of the forces upon the particles in the interior parts of the body of the 

 fluid ; and shows that although the forces at the surface are uni- 

 versally deducible from the general expressions of the forces of the 

 interior parts, yet the converse of this proposition is not universally 

 true, the former not being always deducible from the latter ; a di- 

 stinction which is not attended to in Clairaut's theory. He then in- 

 vestigates the manner in which these two classes of forces are con- 

 nected together; establishes a general theorem on the subject; and 

 proceeds to its application to some of the principal problems, relating 

 to the equilibrium of a homogeneous fluid at liberty, and of which the 

 particles attract one another with forces, first in the inverse duplicate 

 ratio, and secondly in the direct ratio of the distance, at the same 

 time that they are urged by a centrifugal force arising from their re- 

 volution round an axis. The author concludes with some remarks on 

 Maclaurin's demonstration of the equilibrium of the oblate elliptical 

 spheroid ; and on the method of investigation followed in the paper 

 published in the Philosophical Transactions for 1824*. In an Appendix 

 the author subjoins some remarks on the manner in which this sub- 

 ject has been treated by M. Poisson. 



The reading of a paper was then commenced, entitled, " Experi- 

 mental Researches in Electricity;" Eighth Series." By Michael 

 Faraday, Esq., D.C.L., F.R.S. 



June 12. — A paper was read, entitled, "On the Arcs of certain 

 Parabolic Curves." By Henry Fox Talbot, Esq., M.P., F.R.S. 



The general equation to parabolic curves, (namely, nu = v n ; where 

 u is the abscissa and v the ordinate, ) gives for the arc of the curve an 



* A paper by Mr. Ivory also relating to ihis subject will be found in Phil. 

 Mag., vol. lxiii. pp. 339, 347. 



