ON THE RINGS OF SATURN. 115 



forces acting on the rings which favor this hypothesis. For instance, on the assumption 

 that the matter of which the ring is composed is in a solid state, we may compute for 

 any point on its surface the sum of the attractions of the whole ring and of Saturn. The 

 centrifugal force, generated by its rotation, may then be determined from the condition 

 that the particle must remain on the surface. Now in the case of a solid ring, particles 

 on the inner and outer edges must have the same period of rotation. This condition 

 limits the breadth of the ring, for if it be found necessary for the inner and outer edges 

 to have different times of rotation, this can be accomplished only by a division of the 

 ring into two or more parts. In this way Laplace has inferred the necessity of there 

 being several rings. From a more exact analysis, M. Plana, in the Mem. Acad. Turin, 

 Vol. XXIV., concludes that more than one ring is not essential. The data which he as- 

 sumed we now know to have been very wide of the truth, as regards the mass and thick- 

 ness of the ring. 



Bessel's last determination of the mass, derived from the progressive motion of the line 

 of apsides of the satellite Titan, which amounts to a very sensible quantity, makes that 

 assumed by Plana at least thirty times too large. If Bessel's mass be received, the ne- 

 cessity of numerous rings can scarcely be questioned. 



If the density of the ring be the same with that of Saturn, and its matter uniformly 

 distributed, with Bessel's mass = tIt of Saturn's, its thickness, seen from the earth, 

 would only subtend an angle of 2V of a second of arc. It is a confirmation of the mass 

 adopted, that this does not vary more from that derived from observation, than we can 

 attribute without improbability to a difference of density between the ring and Saturn. 

 Sir John Herschel states, Outlines of Astronomy, p. 315, that it cannot be so large as one 

 twentieth of a second. In the Astronomical Journal for January, 1850, I have given as 

 the result of observations with the great refractor at Cambridge, during the disappear- 

 ance of the ring in 1848-49, a thickness not exceeding one hundredth of a second. 

 We cannot suppose the mass to be greater than that assigned by Bessel, without also 

 admitting a density much greater than that of Saturn, the smallest observed thickness 

 already requiring a density more than three times that of the planet. 



In the calculations which follow, I have supposed the mass of the ring not greatly to 

 exceed tIt of Saturn, and its thickness tt of a second. For the other elements I have 

 used Struve's measurements. 



The analysis of the attraction of the ring presents great difficulties. Laplace has 

 taken as an approximation for a very narrow ring the attraction of a cylinder of infinite 

 length, having for its base an ellipse. Plana takes account of the curvature, by assuming 

 the breadth to be very small compared with its radius. But if more than the first term is 



