100 G. P. Bond on the Rings of Saturn. 



"^ 



mass adopted, that this does not vary more from that derived 

 from observation, than we can attribute without improbabiUty 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 observa- 

 tions with the great refractor at Cambridge, during the disappear- 

 ance of the ring in 1848-49, a thickness not exceeding one hun- 

 dredth 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 thick- 

 ness 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 ylg- of Saturn^ and its thickness 

 :^\th of a second. For the other elements I have used Struve's 

 measurements. 



The analysis of the attraction of the ring presents great diffi- 

 culties. Laplace has taken as an approximation for a very nar- 

 row 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 taken into account, the numer- 

 ical calculations become very complicated. These diificulties 

 may in part be avoided by taking account of the form of the sur- 

 face only in the immediate neighborhood of the point attracted. 

 In all the parts distant compared with the thickness, it is sufficient 

 to suppose the whole mass collected in the plane of the center of 

 the rings. This plane, considered as made up of parallel straight 

 lines, attracts the particle by the sum of the attractions of its 

 elements. The attraction of each line parallel to its length, y 

 being its perpendicular distance from the radius joining the at- 

 tracted particle with the center; and r and r' the distances of its 



extremities from the same point, will be 



r 



r 



From which the attraction of a plane surface is easily compu- 

 ted by quadratures. For the ring on the surface of which the 

 attracted particle is, and for the two next adjacent, I have used 

 Laplace's formula, Mecamque Celeste^ vol. ii, [2092]. This as- 

 sumes the figure of the surface to be elliptical ; in the absence of 

 any certain knowledge of its form, this has the recommendation 

 of simpHcity and of satisfying also the conditions of equilibrium- 

 The hypothesis of any other figure would not materially affect 

 the conclusions arrived at^ provided the mass and density be not 

 altered. The numbers thus obtained are only approximations to 

 the truth, but are sufficient for the object in view. 



I 



