contained in the Third Book of the Mécanique Céleste. 85 
three independent co-ordinates. Instead of the equation itself 
we may substitute its fluxion, viz. 
do 
‘ds 
which again amounts to affirming that the resultant of the 
dg de do 
forces ae (dy 2a 
face must be perpendicular to that surface. 
From what has now been shown, it appears that Laplace in 
his investigation has strictly adhered to the received theory 
of fluids. But the algebraic calculus, in its generalizations, is 
apt to overlook distinctions, the neglect of which sometimes 
leads to error and inconclusive reasoning. It never can be 
too often repeated, that analysis is merely an engine of in- 
vestigation, although a very powerful one. All its force and 
all its beauty are derived, as in the ancient geometry, from the 
certainty of the principles on which it proceeds, and from the 
clearness with which it traces their consequences. The mat- 
ter we are considering will furnish an example of a theory 
which is correct in general, but which becomes defective when 
applied in circumstances where a modification is necessary. 
In the foregoing investigation p is the gravity at the level sur- 
face immediately below the external surface; or rather, it is the 
gravity which, according to the principles of the differential 
calculus, is supposed to remain without change from the one 
surface to the other. The force p therefore depends entirely 
on the level surface and the matter within it; p x / x ds is the 
pressure which this force causes by its action on an elementary 
part of the superincumbent stratum. In the investigation of 
Clairaut the pressure mentioned is supposed to be the only 
force which the stratum exerts on the fluid below it; and if 
we allow that it is equable, and likewise that the whole mass 
is in equilibrio, it will follow that the part bounded by the 
level surface will be in equilibrio separately, if the stratum 
above it were taken away or annihilated. But, in the case of 
a planet, there is another force that must be taken into ac- 
count, besides the pressure of the stratum caused by the gra- 
vitation at the level surface: it is the attraction of the stra- 
tum itself upon all the particles within it. The existence of 
this force is a consequence of the law universally prevailing in 
nature, that every particle of matter attracts every other par- 
ticle. Thus, in the case of a planet, the stratum is made to 
press upon the fluid below it by two different forces,—by the 
gravitation at the level surface, and by the attraction which its 
own matter exerts upon the particles within it, Both these 
pressures 
z=0; 
dg dg 
urging a molecule in the external sur- 
