86 Mr. Ivory on the Theory of the Figure of the Planets 
pressures must be equable over the whole level surface, other- 
wise Clairaut’s reasoning will not apply; and as they are pro- 
duced by independent causes, they cannot possibly be included 
in one and the same equation. In the case of a planet it there- 
fore becomes necessary to add to Clairaut’s theory a new con- 
dition, which can only be derived from the figure of the stra- 
tum. If we suppose that the stratum is possessed of such a 
figure as to attract every particle in the inside with equal force 
in opposite directions, it is manifest that the attraction of the 
stratum will not disturb the equilibrium of the fluid below it, 
and consequently that the pressure upon the level surface 
must be equable. Every level stratum being subjected to this 
new condition at the same time that the usual equation of the 
level surface is retained, we may extend Clairaut’s demonstra- 
tion to the case of a planet; and as the two conditions are suf- 
ficient, so we must infer that they are necessary, for the equi- 
librium. Thus are we led to the true conditions requisite to 
the equilibrium of a homogeneous mass of fluid consisting of 
particles that mutually attract one another, which are these 
two; viz. 1st, The resultant of the forces acting at every point 
of the external surface must be perpendicular to that surface ; 
2dly, A stratum of the fluid contained between two level sur- 
faces must attract every particle in the inside with equal forces 
in opposite directions. 
It follows from what has been proved, that the solution of 
Laplace is defective, because one of the conditions that must 
be attended to in the case of a planet is omitted. We have 
no direct evidence that the figure brought out is one of equi- 
librium. Whether the result be exact or not, is accidental, 
and can be known in no other way than by a comparison 
with other solutions derived from unexceptionable principles. 
We can now assign the reason why Maclaurin, and all those 
who supposed an elliptical spheroid, have succeeded in this in- 
vestigation. That condition of equilibrium which has always 
been omitted in the hydrostatical theory, is contained in the 
assumed figure. In a homogeneous ellipsoid the level sur- 
faces are similar to the outer surface; and the author of the 
Principia has proved that a hollow shell of homogeneous 
matter, contained between two similar elliptical surfaces, at- 
tracts a particle in the inside with equal force in opposite di- 
rections. Thus the most difficult and abstruse part of the 
investigation being contained in the very hypothesis assumed, 
the rest of the solution is readily completed by the more ob- 
vious principles of hydrostatics. It deserves to be mentioned 
that of the two requisite conditions, the second, or the one 
which has always been omitted, determines the kind of figure 
without 
