84 Mr. Ivory on the Theory of the Figure of the Planets 
and take the difference, we shall get 
ee ba + Sh dy + So bz =00. 
dz 
As before, put 
sii do? \2 de? 2 d@ 2} 
p=\/ CS, hee 
and it will be easy to prove that the cosines of the angles 
which the lines 2, @y, @z make with / are respectively equal 
; I d 1 d 1 d 
to _ “ SS x hk — oe ; 
‘ Pp dx Pp dy Pp dz 
hence 
k d@ k d@ k d 
=— ey= — Fe —— ee — 
ox Pp dx? J P s dy? p dz 
wherefore, by substituting these values in the foregoing ex- 
pression, we shall obtain 
k d¢\2 dg\2 dg \2 Pp 
— x} a) tha +() kk x ip ep 
Suppose, further, that the level surface is divided into equal 
elementary parts, one of which is ds; then & x ds will repre- 
sent the small mass of fluid contained between the two sur- 
faces upon the base ds; and, p being the resultant of the 
forces that urge every molecule of the small mass, x p x ds 
= ds x 6C will represent its pressure upon the level ‘sur- 
face: wherefore, since C and ds are the same at every point 
of the surface, it follows that the pressure will be equal upon 
all equal spaces of the level surface.. In the very same man- 
ner it may be shown that the second stratum presses equally 
upon the level surface below it; and the same thing is mani- 
festly true of all the successive strata in order. But as the 
number of strata increases, the fluid within them continually de- 
creases in qnantity; ultimately therefore it will be reduced to a 
particle wh:ch exerts no force, and which, being pressed equally 
on all sides, cannot fail to be zn eguélibrio.. Remounting now 
from this central drop to the exterior surface, it is evident that 
the whole mass of fluid, and every part of it bounded by a 
level surface, will be separately zn. equilibrio. ° 
In the foregoing demonstration. we have followed the ideas 
of Clairaut in his excellent work on the figure of the earth*. 
It appears from the reasoning that all the conditions of equi- 
librium are contained in the equation of the external surface, 
viz. @=1C, 
nothing more being necessary than that ¢ be a function of 
Figure de la Terre, orem. vartie 5 XXi. 
* Fig H I a 
three 
