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



Now 2 (dm) = M a constant, 2 (x'dm) = 0, and 2 (x^Sm) is a quantity which 

 increases when the rings are spread out from the mean distance either way, 

 x' being subject only to the restriction 2 (x'dm) = 0. But 2 (x^dm) may 

 increase without the extreme values of x' being increased, provided some other 

 values be increased. 



32. In fact, if we consider the very innermost particle as moving in an 

 ellipse, and at the further apse of its orbit encountering another particle 

 belonging to a larger orbit, we know that the second particle, when at the 

 same distance from the planet, moves the faster. The result is, that the 

 interior satellite will receive a forward impulse at its further apse, and will 

 move in a larger and less eccentric orbit than before. In the same way one 

 of the outermost particles may receive a backward impulse at its nearer apse, 

 and so be made to move in a smaller and less eccentric orbit than before. 

 When we come to deal with collisions among bodies of unknown number, size, 

 and shape, we can no longer trace the mathematical laws of their motion with 

 any distinctness. All we can now do is to collect the results of our investi- 

 gations and to make the best use we can of them in forming an opinion as 

 to the constitution of the actual rings of Saturn which are still in existence 

 and apparently in steady motion, whatever catastrophes may be indicated by 

 the various theories we have attempted. 



33. To find the Loss of Energy due to interned f notion in a broad Fix id 

 Ring, the parts of which revolve about the Planet, each with the velocity of a 

 satellite at the same distance. 



Conceive a fluid, the particles of which move parallel to the axis of x 

 with a velocity u, u being a function of z, then there will be a tangential pres- 

 sure on a plane parallel to xy 



du ., f 



= u. -j- on unit of area 



r dz 



due to the relative sliding of the parts of the fluid over each other. 



In the case of the ring we have 



w = 5*r- 1 . 

 The absolute velocity of any particle is <ar. That of a particle at distance 



(r + 8r) is 



d 



- (wr)8r. 



