Mechanical Illustrations of the Planetary Motions. 331 



equilibrium, by first imagining the ring to be a mere circular 

 circumference, attracted equally in all parts towards a point 

 not coincident with its centre. He then goes on to compute 

 the attraction existing between the centre of the ring and 

 the centre of the planet, and, by a very refined process of in- 

 tegration, he determines that attraction to be negative, or, in 

 other words, that these two centres, instead of attracting, 

 repel each other, and consequently, instead of tending to re- 

 turn to coincidence, will continue to go more apart, until 

 the circumference of the ring touch the surface of the planet. 



It is not necessary for me to produce Laplace's calcula- 

 tion, since I am not going to find any fault with it as far as 

 it goes, and its aspect is such that I am confident its attrac- 

 tions, for this assembly, would turn out to be of a negative 

 kind — somewhat repulsive. All that I need to say is easily 

 appreciated ; and that is, not that the calculation is wrong, 

 but that one of the principal elements is entirely omitted. 

 Of the symbols introduced into the calculation, not one has 

 any reference to the rotation of the ring : it is taken as at 

 rest. How an omission so fatal should have been made on 

 the part of so eminent a mathematician, I cannot explain ; 

 neither am I called upon to account for the circumstance of 

 the oversight not having been detected by subsequent astro- 

 nomers. In the previous part of the demonstration, no doubt, 

 the rotation forms a principal element, but not so in that part 

 which we are now considering. My statement will be found 

 borne out by a reference to La- 

 place's celebrated work, the Meca- /^ 

 nique Celeste, First Part, Book iii., / 

 art. 46. But to save the trouble of f 

 that reference, I will give a short [ 

 sketch of his process. \ /^^V / 



A being the centre of the ring, \ ( B l' j / 



and B that of the planet ; the sym- \. \ / / 



bol S representing the mass of Sa- ^ 



turn ; r t the radius of the ring; m, the angle DAC ; and z, 

 the line AB ;* he then says, — 



* The diagram rests on my own responsibility. There is no diagram for this 

 in Laplace's work. It is presumed that no one will seriously maintain that a 

 ring at rest and a ring in rotation, obey the same laws. 



