332 James Elliot on certain 



Sd7X 



d^ r S 



dzj s/(r 2 + 2r 



*J(r 2 + 2rz cos ZS + z 2 ) 



will be the attraction of Saturn to the ring, decomposed in 

 a direction parallel to z ; the integral being taken fromTS'^O 

 to tJT = the circumference, and the differential being taken 

 with regard to z. 



In all this there is no consideration of the ring's rotation, 

 if I can understand it aright. All that is proved is precisely 

 what we should have anticipated without such proof, viz., 

 that if a planet at rest, surrounded by a ring also at rest, were 

 nearer the one side of the ring than the other, it would be 

 drawn towards that side. Still, although this appears likely 

 beforehand, it is not self-evident ; for, as Sir Isaac Newton 

 has demonstrated, it would not be true in regard to a planet 

 placed within a hollow sphere, which is equally likely. It is 

 very well, therefore, that Laplace has demonstrated it, and 

 set the matter at rest, although, from the omission from the 

 calculation, of any element representing velocity of rotation, 

 the result has no bearing whatever upon the actual case of 

 Saturn's ring,* since, as I think I am prepared to show, the 

 very cause of the stability of equilibrium rests on the omitted 

 consideration. 



Still, Laplace is not easily understood ; and if I have made 

 any mistake regarding his meaning, I shall be glad to be set 

 right-t But, whether or not, it does not affect my result, nor 

 my objection to Laplace's conclusion ; for, according to his 

 own explicit statement, the only force he has computed, is 

 that in the direction parallel to AB. I shall at once assume 

 not only that he is right in that in the case of the ring at 

 rest, but also that the rotation of the ring will not affect that 

 force. I will therefore commence at the point where he has 

 left off; and start with the assumption that there is a repul- 

 sion between the two centres, and, consequently, that, if the 



[ . * It was, however, especially to Saturn's ring that Laplace's investigation 

 was directed, and therefore the rotation was an essential element. 



f I have been censured for presuming to dispute so high an authority as that 

 of Laplace. I am not at all disposed to question Laplace's very high position 

 as a mathematical astronomer ; but the greatest of men are not infallible, and 

 the subject is certainly a fair one for discussion, so long as I assign my reasons, 

 with perfect willingness to retract if proved to be in error. 



