THE EQUILIBRIUM OF INCOMPRESSIBLE FLUIDS. 553 



and as the preceding equations must arise, whatever may be the limits 

 of the integrals, we must necessarily have 



A « »« := -J — a w 

 ax 



Y dm = -J— du) 



dy 



dV 

 dz 



Ij am = —j — d w , 



/, , . dm \ dP ^^ dP ^ dP 



or (^by makmg J^ = f j ^"^ = ? X, ^ = f Y,~ = p Z : 



f, which expresses the ratio of the mass dm to the volume of dcu, is 

 but the expression of the density. 



Let us now particularly consider a homogeneous liquid, the surface 

 of which is entirely free and suffers no external pressure, we shall have 

 at first for all the elements of tiie liquid, whatever dx, dy, dz may be, 



6?j9 = f (X rfa; + Y rfy + Z rfz), 

 and then, for the surface 



= XcZa! + Xdy + Zdz. 



The last equation shows that the resultant of X, Y, Z is normal to 

 the surface of the liquid. It is obvious that in the expression X. dx 

 + Y dy + Zdz, the differentials d x, dy, d z belong to the passage from 

 a molecule of the surface to another molecule situated also at the sur- 

 face ; but if we passed from a molecule at the surface to a molecule si- 

 tuated in the interior, we should have Xdx + Ydy + ZdzyO; 

 which requires that the resultant of X, Y, Z should be directed towards 

 the interior of the liquid mass. Thus it appears necessary for the equi- 

 librium of a homogeneous liquid mass, that the differential Xd x + Y dy 

 + Z d z should be exact for all the points of the mass, that the result- 

 ant of the forces X, Y, Z should be normal to the surface in all the 

 elements of the surface, and that it should always act in the direction 

 of the interior of the liquid. If these conditions are not fulfilled, it 

 might be supposed that the liquid mass could never remain in equilibrio. 

 I have, however, observed a case in which, although the last of these 

 conditions is not satisfied, yet there is certainly an equilibrium. 



Let us siqjpose that the liquid forms a spherical shell of any given 

 thickness, and that each of its molecules is attracted towards the centre 

 by a foi-ce proportioned to a function of the distance between the 



