116 ON THE RINGS OF SATURN. 



taken into account, the numerical calculations become very complicated. These diffi- 

 culties may in part be avoided by taking account of the form of the surface 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 

 centre 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 centre, and r and r' the distances of its extremities from the 



the same point, will be ( \(^y- 



From which the attraction of a plane surface is easily computed 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, Mecanique Celeste, Vol. II. [2092]. This assumes the 

 figure of the surface to be elliptical ; in the absence of any certain knowledge of its form, 

 this has the recommendation of simplicity, and of satisfying also, to some extent, 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. 



If we adopt for unity the radius of the outer edge of the outer ring, we have from ob- 

 servation the thickness of the ring = 2 fe < 9-00. Let r and r' be the radii of the inner 

 and outer edges, and i the interval between the two adjacent rings, 



n = ^— . 2 a = r - r. 

 Any intervals permanently existing so large as one half, or even one third, of that 

 usually seen, could not escape observation. Moreover, if the subdivisions are numerous, 

 the width of the intervals must be proportionally diminished, because the whole area oc- 

 cupied by them goes to diminish the amount of light reflected, and to increase the den- 

 sity of each ring, both of which are already large. The light of the ring being sensibly 

 brighter than that from an equal area on the ball, it is not probable that any considerable 

 part of the light of the sun is transmitted through intervals. And to preserve the same 

 mass, if the intervals are large, the matter must be compressed, as it is not allowable to 

 give a thickness greater than is indicated by observation. To avoid the hypothesis of a 

 reflective power, and a density greater than we are warranted in assuming, we must, 

 therefore, consider the intervals to be very narrow. We may take, then, the width of all 

 but the known interval as certainly less than 0.01, which is one half of the width of the 

 known interval. From the blackness of the shadow of the ring upon the ball, which 

 would be diminished in intensity were a considerable part of the sun's rays transmitted, 



