ON THE STABILITY OP THE MOTION OF SATURN'S KINGS. 303 



ring near the given point, fi will be a periodic function of 0, and may there- 

 fore be expanded by Fourier's theorem in the series, 



/u, = {l + 2/cos0 + f#cos20 + f/ism20+2icos(30 + a) + &c.} (21), 



i.TT(.t 



where f, g, h, &c. are arbitrary coefficients, and R is the mass of the ring. 



(l) The moment of the ring about the diameter perpendicular to the 

 prime radius is 



-f' 



Jo 



cos OdO = Raf, 



therefore the distance of the centre of gravity from the centre of the ring, 



r = a/ 



(2) The radius of gyration of the ring about its centre in its own plane 

 is evidently the radius of the ring =a, but if k be that about the centre of 

 gravity, we have 



(3) The potential at any point is found by dividing the mass of each 

 element by its distance from the given point, and integrating over the whole 

 mass. 



Let the given point be near the centre of the ring, and let its position be 

 denned by the co-ordinates r' and |>, of which r' is small compared with a. 



The distance (p) between this point and a point in the ring is 

 ^ = i{l + 



r 

 The other terms contain powers of - - higher than the second. 



We have now to determine the value of the integral, 



o P 



and ha multiplying the terms of (p.) by those of (-) , we need retain only 

 those which contain constant quantities, for all those which contain sines or 





