136 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
The method of solution we have here followed may be applied to all pro- 
blems concerning the equilibrium of a mass of fluid, when it is possible to form 
the equation of the outer surface ; that is, when the forces in action at all the 
points of the outer surface are the same functions of the coordinates of those 
points, whatever geometrical figure the mass may be supposed to assume. This 
in reality comprehends every question that has hitherto occurred ; and, as the 
conditions which we have laid down are necessary and sufficient for the equili- 
brium in every hypothesis of the forces that can be imagined, we shall not 
enter into any further discussion of this point. 
9. The preceding analysis, by which we have investigated the figure of equi- 
librium of a homogeneous planet is direct and unexceptionable in point of 
rigour. It seems hardly possible to express simply in algebraic language, all 
the forces that urge the interior particles of the fluid ; and this makes it neces- 
sary to have recourse to peculiar modes of reasoning for determining the figure 
of equilibrium. The problem, being one of great importance and difficulty, 
which has much engaged the attention of geometers, and which requires for its 
solution principles different from those that have so long passed current with- 
out suspicion of their accuracy, it may not be improper to add another investi- 
gation of it by a process of reasoning very different from the foregoing. 
Second investigation. 
We shall begin with laying down the following lemma. If a mass of homo- 
geneous fluid, consisting of particles which attract one another inversely as the 
square of the distance, be in equilibrium when it revolves with a certain angu- 
lar velocity about an axis ; any other mass of the same fluid, the particles at- 
tracting by the same law, will be in equilibrium, if it have a similar figure, and 
revolve with the same rotatory motion about an axis similarly placed. 
Take any two particles similarly placed in the two bodies, and having the 
same proportion to one another as the whole masses ; it is proved in the Prin- 
cipia of Newton, and in the works of other authors, that the resultants of the 
attractive forces acting upon the particles, have similar directions, and are pro- 
portional to the linear dimensions of the two bodies. Further, the centrifugal 
forces urging the two bodies to recede from the axes of rotation, are propor- 
tional to the respective distances from the axes, that is, to the linear dimen- 
