Analijsis of the Mecanique Celeste of M. La Place. £59 



which animate it ; and it is according to the condition of 

 this equihijnum that he determines the figure of the rincs. 

 To obtain it, he conceives each ring as engendered by The 

 revohition of a ch)scd tigutc, snch as the elhpse moved per- 

 pendicularly to its plane about the centre of Saturn placed 

 on the prolongation of the axis of this figure. Introducing; 

 these circumstances in the equation of the second order of 

 partial diflerenccs relative to the attraction of spheroids, 

 and supposinir the dimensions of the ring to be very small 

 with regard to its distance from Saturn's centre, there arises 

 an integral equation, which is the same as if the annular 

 surface were a cylinder of an infinite length : and in fact 

 we see that this case is very nearly that of the rinc: w hen 

 the attracting point is near its surface. But, as this first ap- 

 proximation is not in general sufficient, ihe author gives the 

 means of obtaining others more and more exact ; and he 

 shows, that to obtain them it suP.ices to know the attrac- 

 tions of the rings on points placed in the prolongation of 

 t!)c axis of their generating figure. Considering in parti- 

 cular the case where this figure is an ellipsis, he gives the 

 values of these attractions, as well on a point distant from 

 the rings as on a point on their surface. 



He then supposes the ring to be a fluid homojreneous 

 mass, and the generating curve to be an ellipsis. The ge- 

 neral equation of equilibrium shows in this hvpothesis the 

 motion of rotation of> the ring, and the cUipticity of the 

 generating curve ; he deduces from it again the limits of 

 the ratio of the mean density of Saturn to thai of the ring; 

 and at last obtains this remarkable result, that the motion 

 of the ring is the same as that of a salellite as far distant 

 from the centre of Saturn as the cenVre of the generatini>' 

 figure is from it ; which exactly conforms with the obser- 

 vations, lie then shows that the preceding theory would 

 still subsist if the generating ellipse varied its , magnitude 

 and position throughout the whole extent of the circum- 

 Icrence of the ring, which might thus be supposed of an 

 unequal size in it* difFercut parts, as appears to take place 

 in nature. Lastly, he demonstrates that these inequalities 

 are necessary to maintain the ring in equilibrium round 

 Saturn: to prove it, he supposes the ring to he a circular 

 line whose plane passes by ihe centre of Saturn, but with- 

 out the two centres coinciding : and he shows that then the 

 centre of Saturn will always repel the centre ol the ring; so 

 that, whatever may be the motion of this second centre 

 round the first, the curve which it describes would be con- 

 vex lowaids Saturn : it would finibli thercfoie by receding 

 li '■2 more 



