120 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
therefore, that the difference between the conditions of equilibrium hitherto 
universally adopted, and those laid down above, amounts to this : according 
to the former it is required that the expressions of the accelerating forces be 
the same functions of the co-ordinates at every point of the outer surface, this 
being all that is necessary for forming the differential equation of that surface; 
according to the latter, the forces will not balance the pressure, and the laws 
of equilibrium will not be fulfilled unless the forces be the same functions of 
the co-ordinates at every point whether situated in the outer surface, or in the 
interior part of the mass. 
If a homogeneous fluid, of which the particles are urged by accelerating 
forces be in equilibrium, all that is required by Clairaut’s theory will un- 
doubtedly be fulfilled ; but the converse of this cannot be affirmed. It is no 
where proved generally by unexceptionable arguments, and indeed no proof 
can possibly be given, that the forces in the interior parts of the fluid will 
balance the pressure, merely because the resultant of the forces in action at 
the outer surface is perpendicular to that surface. All the attempts that have 
been made to demonstrate this point, tacitly assume that the expression of the 
forces is the same at the surface and in all the interior parts ; which is not uni- 
versally true. 
In a very extensive class of problems the difference between the two ways 
of laying down the conditions of equilibrium disappears. This will happen 
when the accelerating forces are independent of the figure of the fluid, as will 
be the case if the particles exert no action on one another by attraction or 
repulsion. In such problems the forces impressed upon every particle, what- 
ever be its situation, and whatever be the figure of the fluid, are by the hypo- 
thesis, the same given functions of the co-ordinates. The figure of equilibrium 
will be the same whether, following Clairaut, we obtain the equation of the 
outer surface by means of the forces in action at that surface, or, making use 
of the property that the pressure vanishes at all the points where the fluid is at 
liberty, we deduce the same equation from the pressure that prevails generally 
throughout the mass. 
But Clairaut’s theory cannot be extended to the solution of other problems 
than those of which we have been speaking. In no other cases is it evident 
without inquiry that the proposed accelerating forces urging a particle, are, in 
