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



cated by the solution would cause a small periodic variation, or a total 

 derangement of the motion." 



The question may be made to depend upon the conditions of a maximum 

 or a minimum of a function of many variables, but the theory of the testa 

 for distinguishing maxima from minima by the Calculus of Variations becomes 

 so intricate when applied to functions of several variables, that I think it doubt- 

 ful whether the physical or the abstract problem will be first solved. 



PART I. 



ON THE MOTION OF A RIGID BODY OF ANY FORM ABOUT A SPHERE. 



WE confine our attention for the present to the motion in the plane of 

 reference, as the interest of our problem belongs to the character of this motion, 

 and not to the librations, if any, from this plane. 



Let S (Fig. 2) be the centre of gravity of the sphere, which we may call 

 Saturn, and R that of the rigid body, which we may call the Ring. Join RS, 

 and divide it in G so that 



SG : GR :: R : S, 



\ 



R and S being the masses of the Ring and Saturn respectively. 



Then G will be the centre of gravity of the system, and its position will 

 be unaffected by any mutual action between the parts of the system. Assume G 

 as the point to which the motions of the system are to be referred. Draw GA 

 in a direction fixed in space. 



Let AGR = 6, and SR = r, 



o D 



then GR = ~S+R T> and GS= ~S+R r> 



so that the positions of S and R are now determined. 



Let BRR be a straight line through R, fixed with respect to the substance 

 of the ring, and let BRK=<j>. 



