contained in the Third Book of the Mécanique Céleste. 83 
tuated in the surface or any where in the interior of the fluid, is 
urged by the forces sf. re = 
ordinates and tending to shorten them, it being always under- 
stood that the co-ordinates of the molecule are to be substi- 
tuted in the expressions of the forces. 
In order to demonstrate this proposition take the differential 
of the equation of the surface ; then 
in the direction of the co- 
dQ dg eg, = 
ee dx + ay dy + Te dz= 
now 4, oe : = are the forces which act upon a molecule 
in the surface at the point of which a, y, z are the co-ordinates ; 
and if we put 
x i <aBy ynesey 
Pir dx ) a ( a: ds 
then p, which is the resultant of the partial forces, will repre- 
sent the gravity at the exterior surface; and it is easy to de- 
duce from the differential equation that the direction of p will 
be perpendicular to that surface. Suppose that the constant 
- quantity C in the equation of the fluid’s surface decreases by 
a small variation, then beh Chea 
dq 
dy 
] 
which will be the equation of a surface in the interior of the 
mass indefinitely near the outer surface: and since the forces 
which urge every molecule of the fluid are expressed by the 
same functions of the co-ordinates of the molecule, it follows 
that the resultant of the forces acting at any point of this new 
surface will be perpendicular to it, for the same reason that 
the like resultant is perpendicular to the outer surface. And 
as we may conceive that C decreases by indefinitely small 
gradations till it is entirely exhausted, it is evident that the 
whole fluid mass may be supposed to be divided into thin 
strata separated from one another by surfaces perpendicular 
to the forces which urge the molecules contained in them. 
Clairaut has called such surfaces Couches de niveau, or level 
surfaces, from the property which they possess in common of 
being perpendicular to the direction of gravity. Again, let & 
denote the perpendicular distance between the outer surface 
of the fluid and the level surface immediately below it; and 
iet x, y, x be the co-ordinates of the extremity of / in the first 
surface, and « =8a, y — dy, 2 —%3% the co-ordinates of the 
other extremity in the other surface; then if we substitute the 
respective co-ordinates in the equations 
g=C 
g¢=C—8C, 
and 
