ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 363 



To exhibit the movements of the satellites, I have made an arrangement 

 Jy which 36 little ivory balls are made to go through the motions belonging 

 to the first or fourth series of waves. (Figs. 7, 8.) 



The instrument stands on a pillar A, in the upper part of which turns 

 the cranked axle (7(7. On the parallel parts of this axle are placed two wheels, 

 RR and TT, each of which has 36 holes at equal distances in a circle near 

 its circumference. The two circles are connected by 36 small cranks of the 

 form KK, the extremities of which turn in the corresponding holes of the two 

 wheels. That axle of the crank K which passes through the hole in the wheel 

 S is bored, so as to hold the end of the bent wire which carries the satellite S. 

 This wire may be turned in the hole so as to place the bent part carrying 

 the satellite at any angle with the crank. A pin P, which passes through the 

 top of the pillar, serves to prevent the cranked axle from turning ; and a pin Q, 

 passing through the pillar horizontally, may be made to fix the wheel R, by 

 inserting it in a hole in one of the spokes of that wheel. There is also a 

 handle //, which is in one piece with the wheel T, and serves to turn the axle. 



Now suppose the pin P taken out, so as to allow the cranked axle to 

 turn, and the pin Q inserted in its. hole, so as to prevent the wheel R from 

 revolving ; then if the crank C be turned by means of the handle //, the 

 wheel T will have its centre carried round in a vertical circle, but will remain 

 parallel to itself during the whole motion, so that every point in its plane will 

 describe an equal circle, and all the cranks K will be made to revolve exactly 

 as the large crank C does. Each satellite will therefore revolve in a small 

 circular orbit, in the same time with the handle H, but the position of each 

 satellite in that orbit may be arranged as we please, according as we turn the 

 wire which supports it in the end of the crank. 



In fig. 8, which gives a front view of the instrument, the satellites are so 

 placed that each is turned 60 further round in its socket than the one behind 

 it. As there are 36 satellites, this process will bring us back to our starting- 

 point after six revolutions of the direction of the arm of the satellite ; and 

 therefore as we have gone round the ring once in the same direction, the arm 

 of the satellite will have overtaken the radius of the ring five times. 



Hence there will be five places where the satellites are beyond their mean 

 distance from the centre of the ring, and five where they are within it, so 

 that we have here a series of five undulations round the circumference of the 



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