40 



STATEMENT AND EXPOSITION OP 



(C) representing (pH^ in the figure, i.e. the distance of the centre of gyration from 



the centre of force, and B and r, respectively, 

 the radii of tlie edges of the ring, so that we 

 have 



2 2 2 



Now the like being also true of the half- 

 rings, with their centres of gyration at /i'2 and 

 i^4, respectively; we shall also have 



<pi22 = l{<pR^ + cpR,) ; and 



_2 2 2 



cpR^ = ^{<pRs + <pRi); 

 from which, by substitution and reduction, we 



shall obtain 



2 2 2 



^2^3= ^(<pR., + cpR,); 



in which the centres of gyration of the halfrmgs respectively, take the places of 

 the edges of the whole ring. 



(55) The supposition here throughout has been that all the material was homo- 

 geneous. But as the " abandoned" rings, or rmg-like masses, would increase in 

 density inward, the centre of gyration for each half-ring, as well as that of the 

 whole ring, would also, therefore, be within that assigned by the formula. 



Nevertheless it would seem that this would affect, or rather has affected, the 

 several quantities, proportionally. 



Accordingly, we find that the mass of the system of the inner bright rings of 

 Saturn is considerably greater than the iwiss of the system of the outer bright 

 rings; yet the other condition here in question is fulfilled. 

 For the centre of gyration of the outer bright rings, [Table (C) in 



(18)], is at the distance 2.1165. ' 



And the centre of gyration of hoth systems of the bright rings, as 

 obtained independently by the general formula, is at distance . 1.9090. 

 And that of the system of the mner bright rings is at . . . 1.7097. 

 Now the sum of the squares of the first and last of these numbers is 7.16399197; 



' and i of the same = e3.58199593 + 



And the square of the intermediate number, 1.9090, = 3.64428100; 



showing a very close correspondence with the formula. 



Accepting, then, this result as an induction, we shall find, on trial, in the same 

 way, a semblance of a ring-like form of the " abandoned" masses, apparent, even 

 in the case of the Earth and Venus. 



For the sum of the squares of their mean distances [as those distances 

 are given in the column oi Law in Table (B) in (14)] is . . . 1.51928 



And, (C) being distance of the centre of gyration, 



and \ sum 



: {cf 



0.75964 

 0.78616 



