18() MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



*198. Newton s Revolving Orbit. Suppose that the form 

 of the tube in Article 197 is a free path under a central force to 0. 

 Let the tube turn about with an angular velocity cf&amp;gt; which is 

 always equal to n&, where n is constant, and 6 is the angular 

 velocity of the radius vector in the free path when the particle is 

 at (r, 6). Then the path traced out by the particle is a free path 

 under the original central force and an additional central force 

 which varies inversely as the cube of the distance. 



Let / be the central acceleration in the free path, and ^h the 

 rate of description of areas. Then we are given 



f-f* /. 



r*6 = h 



Now, in the tube c/&amp;gt; = n0, so that 



and r-r(0+ tf&amp;gt;) 2 = -/- rfr (2n + n~), 



= -/-~(2n + n ). 



Hence the path traced out by the particle in the revolving- 

 tube is a free path with a central acceleration to made up of two 

 terms, one of them being/ and the other being inversely propor 

 tional to r 3 . 



This result may be stated in another form as follows : Rela 

 tively to a certain frame a particle describes a central orbit about 

 the origin with central acceleration /; if a second frame with the 

 same origin rotates about the origin relatively to the first frame, 

 with an angular velocity always the same multiple, of that of the 

 radius vector in the said central orbit, the path of the particle 

 relatively to the second frame is again a central orbit with the 

 central acceleration increased by an amount inversely proportional 

 to the cube of the distance. 



*199. Examples. 



1. A particle moves in a tube in the form of an equiangular spiral which 

 rotates uniformly about the pole, and is under the action of a central force 

 to the pole of the spiral. Prove that if there is no pressure on the tube the 

 central force at distance r must be of the form Ar + Br~ 3 , where A and B 

 are constants. 



2. Prove that motion which, relatively to any frame, can be described 

 as motion in a central orbit with acceleration ^/(distance) 3 towards the origin 



