ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 375 



of the whole force of attraction of S on the rigid body. Then since this force is in the 

 line through S, its moment round G is 



SYx-SXy; 



the components of the forces on the moving body being reckoned as positive when they 

 tend to diminish x and y respectively. Hence if k denote the radius of gyration of the 

 body round G, and if <f> denote the angle which OG makes with SX (i.e. the angle OOK), 

 the equations of motion are, 



In the first place we see that one integral of these equations is 



This is the "equation of angular momentum." 



In considering whether the motion round S with velocity eo when coincides with 

 8 is stable or unstable, we must find whether every possible motion with the same 

 " angular momentum " round 8 is such that it will never bring to more than an infinitely 

 small distance from <S : that is to say, we must find whether, for every possible solution 

 iu which H = M (a 1 + k*) a>, and for which the co-ordinates of are infinitely small at one 

 time, these co-ordinates remain infinitely small. Let these values at time t be denoted 

 thus: /82V = and N0=ij; let OG be at first infinitely nearly parallel to OX, i.e. let </> 

 be infinitely small (the full solution will tell us whether or not $ remains infinitely small) ; 

 then, as long as $ is infinitely small, we have 



x==a+ f , y = r) + a<l>, 

 and the equations of motion have the forms 



and we may write the equation of angular momentum instead of the third equation, 



If now we suppose and i) to be infinitely small, the last of these equations becomes 



> + 2 & ,af+a=0 ...................................... (a). 



