ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 339 



acted on by any external forces, the pressures are due mainly to the component 

 of the attraction which is perpendicular to the plane of the stratum. It is 

 easy to shew that a fluid acted on by such a force will tend to assume a 

 position of equilibrium, in which its free surface is plane ; and that any irregu- 

 larities will tend to equalise themselves, so that the plane surface will be one 

 of stable equilibrium. 



It is also evident, that if we consider only that part of the attraction 

 which is parallel to the plane of the stratum, we shall find it always directed 

 towards the thicker parts, so that the effect of this force is to draw the fluid 

 from thinner to thicker parts, and so to increase irregularities and destroy 

 equilibrium. 



The normal attraction therefore tends to preserve the stability of equilibrium, 

 while the tangential attraction tends to render equilibrium unstable. 



According to the nature of the irregularities one or other of these forces 

 will prevail, so that if the extent of the irregularities is small, the normal 

 forces will ensure stability, while, if the inequalities cover much space, the 

 tangential forces will render equilibrium unstable, and break up the stratum into 

 l>eads. 



To fix our ideas, let us conceive the irregularities of the stratum split up 

 into the form of a number of systems of waves superposed on one another, 

 then, by what we have just said, it appears, that very short waves will disap- 

 pear of themselves, and be consistent with stability, while very long waves will 

 tend to increase in height, and will destroy the form of the stratum. 



In order to determine the law according to which these opposite effects 

 take place, we must subject the case to mathematical investigation. 



Let us suppose the fluid incompressible, and of the density k, and let it 

 be originally contained between two parallel planes, at distances +c and c 

 from that of (xy), and extending to infinity. Let us next conceive a series of 

 imaginary planes, parallel to the plane of (yz), to be plunged into the fluid 

 stratum at infinitesimal distances from one another, so as to divide the fluid 

 into imaginary slices perpendicular to the plane of the stratum. 



Next let these planes be displaced parallel to the axis of x according to this 

 law that if a: be the original distance of the plane from the origin, and its 

 displacement in the direction of x, 



g=A cos rax (81). 



432 



