12 



STATEMENT AND EXPOSITION OF 



SATELLITE SYSTEMS, 



Fig. 1 



System of Saturn. 



(15) In the System of Saturn we find again three ratios; all of them fractional 

 powers of one anothei", and one of these, like the special one in the Planetary 

 System, the square root of another. 



The rings, both bright and dusky, have also their places in the satellite series, 



with the condition always understood, that the 

 distance of any ring from Saturn's centre is to 

 be measured from that ring's oion centre of gyra- 

 tion. 



(16) Now the centre of gyration of an indefi- 

 nitely thin ring, and one which has, in efi'ect, a 

 uniform density and thinness, this centre, has 

 itself special relations which it will be well to 

 notice. 



For let R be the radius of the outer edge of 

 the ring, C the distance of the centre of gyration 

 from Saturn's centre (or from the common centre 

 of all the circles in question), and r the radius 

 of the inner edge of the ring. 

 Then, we have 



C- \ ^' 



-2r'- 



or, 



C: 





E'-r" 



That is 



I (^^+OCR^-r^. 



C=yi(^B'+r') .... (A). 

 But now, if the ring be supposed to be so divided by the circumference of a 

 circle concentric with the edges of the ring, that the two portions thus obtained 

 shall be equal in area, and the radius of this bisecting circumference be x ; then 

 the expressions for the two portions of the ring will be equivalent to one another, 

 and so we shall have 



7i{R^ — x^) = 7t(cc^ — r'^) ; whence 

 R^ ~ x^ = x" — r- ; and 

 i2^ -|- r^ — 2a;^ ; whence 

 x^ = l(i22 ^ r) ; and 

 X = ^i(m'^?) (B). 



Dr. Olinthus Gregory's Mechanics, 4th edition, Art. 312, Ex. III. 



