CERTAIN HARMONIES OP THE S(5LAR SYSTEM. 39 



Vn'of [(U)1^]^l nOl (-3)1 



m'"^/"" of W 



Lastly, in the Sjstem of Saturn, m being the mass of tlie outer, and m' that of 

 the inner bright system of rings ; we shall have for the ratio of the moments of 

 (simultaneous) rotation — 



m X a^ of outer rings , 1 j^mm fA\ 



m' X a''^ of intwr rings 



the rings being respectively referred, each to its centre of gyration [obtained as in 



(16)]. 



[Then, since the rings in Table (C) in (18), have their places as satellites; if the 

 periodic times of the rings referred to their centres of gyration agree with Kej)ler's 

 Sd Law, and so actual velocities are as ai to a'i, and hence their 2d pow^ers as a to 

 a'; we shall have for the ratio of the moments of rotation of the existing and 

 turning ringrs 



m' X a' of inner r 

 m X <-( of outer rvi 



= 1.0752 



^•] 



There is a very close resemblance between ratios (1) and (4).^ Were, then, 

 those ancient masses compared in (1), ring-liJce in form; and did the masses, with 

 nearly equal moments of {simultaneous) rotation, go round the central body 

 together'? 



If, in an ancient state, they were par^s of the atmospheres of their primary and 

 central body, in every case; then they did go round together. But, whether we 

 admit any part of that hypothesis, or else reject any portion, or all of it; the 

 RATIOS REMAIN, and seemingly without that hypothesis, they remain unaccounted 

 for. 



There is yet another aspect of the matter, and that is — that the rings or shells, 

 etc., separated about the time ivhe^i tlie moments in question became nearly equal. 



Application of other Conditions appertaining to the ring-like Form. What succeeded 

 these. — Position of great Planets, and of largest Satellites. 



(54) It has, (16), been shown that the centre of gyration of a homogeneous ring is 

 in the circumference in which the mass of the ring is bisected; and that thus, Ave 

 have 



(f7)==i(i2^ + r=) 

 V 



' Ratios (2) and (3) are consistent with the supposition in (43), that the material of Saturn was 

 gathered in part from the interior half-planet, now missing (the values T^ E^nd §2 being dependent on 

 that) ; but they did not seem to be of such importance as to require their admission as Coincidence 

 \^ih of the series exhibited in (43) and (45). 



^ Though it should not be overlooked that ratio (4) is that existing in a satellite system, which 

 is here compared with those found in the system of the primary planets. 



