ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 365 



In this way we can exhibit the motions of the satellites in the first and 

 fourth waves. In reality they ought to move in ellipses, the major axes being 

 twice the minor, whereas in the machine they move in circles : but the character 

 of the motion is the same, though the form of the orbit is different. 



We may now show these motions of the satellites among each other, com- 

 bined with the motion of rotation of the whole ring. For this purpose we 

 put in the pin P, so as to prevent the crank axle from turning, and take 

 out the pin Q so as to allow the wheel R to turn. If we then turn the 

 wheel T, all the small cranks will remain parallel to the fixed crank, and the 

 wheel R will revolve at the same rate as T. The arm of each satellite will 

 continue parallel to itself during the motion, so that the satellite will describe 

 a circle whose centre is at a distance from the centre of R, equal to the arm 

 of the satellite, and measured in the same direction. In our theory of real 

 satellites, each moves in an elh'pse, having the central body in its focus, but 

 this motion in an eccentric circle is sufficiently near for illustration. The 

 motion of the waves relative to the ring is the same as before. The waves 

 of the first kind travel faster than the ring itself, and overtake the satellites, 

 those of the fourth kind travel slower, and are overtaken by them. 



In fig. 11 we have an exaggerated representation of a ring of twelve satel- 

 lites affected by a wave of the fourth kind where m = 2. The satellites here lie in 

 an elh'pse at any given instant, and as each moves round in its circle about 

 its mean position, the ellipse also moves round in the same direction with half 

 their angular velocity. In the figure the dotted line represents the position of 

 the ellipse when each satellite has moved fonvard into the position represented 

 by a dot. 



Fig. 13 represents a wave of the first kind where m=2. The satellites at 

 any instant lie in an epitrochoid, which, as the satellites revolve about their 

 mean positions, revolves in the opposite direction with half their angular velocity, 

 so that when the satellites come into the positions represented by the dots, 

 the curve in which they lie turns round hi the opposite direction and forms the 

 dotted curve. 



In fig. 9 we have the same case as in fig. 13, only that the absolute orbits 

 of the satellites in space are given, instead of their orbits about their mean 

 positions in the ring. Here each moves about the central body in an eccentric 



