CERTAIN HARMONIES OF THE SOLAR SYSTEM, 13 



The value of x in equation (B) is the same with that of C in equation (A). 

 Hence 



C=x; 



or ilie centre of gyration is in the drcumfereiice of a circle concentric loith the edges of 

 the ring, and bisecting its area. 



And a cylindrical surface having this bisecting circle for one of its edges, and 

 cutting perpendicularly through a ring formed like that of Saturn, w^ould (density 

 uniform) also bisect the volume of the ring, and also would bisect the material of 

 the ring; and the value of C, the centre of gyration of this ring of sensible thick- 

 ness, would not be affected by these new circumstances ; the indefinitely thin ring 

 being the plane of rotation on which the other might be projected/ 



(17) The equation for the centre of gyration of any two equal masses will take 

 the same form as that of Eq. (B), with the condition, however, that R and r shall 

 respectively denote the radii of gyration of those masses. Indicating these radii 

 then by R' and r', and the masses (equivalent or not) by ilf and m ; and then (since 

 velocities are as radii of simultaneous rotation) the general formula will be thus 

 expressed : — 



I MR' + 7?ir'^ p, _ 



which, when M=m, is reduced to 



C=^i{E' + r'') (C)', 



so that when the equivalent masses are both rings, the one wholly clasping the 

 other, like the two halves of the ring in question, the position of the centre of 

 gyration may be obtained by a similar process, whether the ^ sum of the squares 

 under the radicle be that of those quantities representing the radii of outer and 

 inner perimiters of the wliole ring, as in Eq. (A) ; or the radii of gyration of the 

 respective halves, as in Eq. (C)'. 



' This property of the centre of gyration of a ring like those of Saturn, as well as of the indefinitely 

 thin ring, has about it a species of mathematical elegance. I know not whether the enunciation of it 

 is new ; but the correspondence of the position assigned by it with that of the division between the 

 bright ring systems of Saturn, is a curious, if not an interesting one. [See Article (19).] 



