ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 297 



This determines the angular position of the ring, so that from the values 

 of r, 6, and <jt the configuration of the system may be deduced, as far as relates 

 to the plane of reference. 



We have next to determine the forces which act between the ring and 

 the sphere, and this we shall do by means of the potential function due to 

 the ring, which we shall call V. 



The value of V for any point of space S, depends on its position relatively 

 to the ring, and it is found from the equation 



F /dm 



W 4 \ *- 



where dm is an element of the mass of the ring, and r r is the distance of that 

 element from the given point, and the summation is extended over every element 

 of mass belonging to the ring. V will then depend entirely upon the position 

 of the point S relatively to the ring, and may be expressed as a function 

 of r, the distance of S from R, the centre of gravity of the ring, and <j>, the 

 angle which the line SR makes with the line RB, fixed in the ring. 



A particle P, placed at S, will, by the theory of potentials, experience a 



moving force P -j- in the direction which tends to increase r, and P j-r 



dr r d(f> 



in a tangential direction, tending to increase <f). 



Now we know that the attraction of a sphere is the same as that of 

 a particle of equal mass placed at its centre. The forces acting between the 



dV 

 sphere and the ring are therefore S -j tending to increase r, and a tangential 



1 dV 

 force S jr > applied at S tending to increase <f>. In estimating the effect of 



1 dV 

 this latter force on the ring, we must resolve it into a tangential force S - -j-r 



dV 

 acting at R, and a couple S -j-r tending to increase <. 



We are now able to form the equations of motion for the planet and the 

 ring. 



VOL. I. 38 



