32 Mr. Ivory on the Theory of the Figure of the Planets 
is not correctly obtained. In a legitimate investigation we 
must first know all the conditions of equilibrium: we must 
then demonstrate that in particular circumstances some of 
them to a certain extent become unnecessary ; and having thus 
obtained sure principles to proceed upon, we may employ 
mathematical reasoning and the operations of analysis to com- 
plete our purpose. A calculation cannot be unexceptionable, 
even although it lead to a result not erroneous, when a neces- 
sary principle has escaped the penetration of the analyst. We 
may add, that a theory can never be reduced to the utmost sim- 
plicity of which it is capable, unless the physical principles 
are completely separated from the mathematical processes. 
These observations will help to explain the purpose intended 
by the present discussion. It would be ridiculous and a want 
of common sense to object on slight grounds to any thing 
sanctioned by the name of Laplace, or to detract from a re- 
putation placed on such solid foundations, and which will al- 
ways derive part of its lustre from the theory to which our 
present attention is directed. But in an intricate and difficult 
investigation, every bay and creek that can possibly lead to 
error must be explored, before the right track is discovered, 
and before we can arrive at a successful termination. ‘The 
researches of Maclaurin and Clairaut were occasioned by the 
speculations of Newton; the labours of Legendre and Laplace 
were intended to perfect those of their predecessors; and, if 
some steps still remain to be made, there is a field fairly open 
to future inquirers. 
It follows from what has been shown in the last number of 
this Journal, that, in the theory of Laplace, the equation at the 
surface of the spheroid is always true when the molecules, or 
small masses of matter on the surface of the sphere, are placed 
at a distance from the assumed point. In these circumstances 
the thickness of the molecules may vary in any manner with- 
out being subject to the law of continuity. But the equation 
cannot be true for molecules indefinitely near the assumed 
point, unless their thickness be restricted to a certain class of 
functions. In his later writings Laplace, supposing the law of 
attraction to be as in nature, has limited the equation to the 
case when the thickness of the stratum near the point of con- 
tact of the sphere and spheroid decreases as the square of the 
distance from that point. With this limitation the equation is 
no doubt rigorously demonstrated: but we are still left in the 
same uncertainty as before; since we are not informed what 
kind of functions is comprehended under the hypothesis as- 
sumed. When this point is inquired into, it turns out that 
the theorem is now too much restricted for the use to be made 
of 
