64 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



and the complete primitive is (by Article 47) 



v=A cos (6 a), 

 where A and a are arbitrary constants. If we put now e for Al we have 



so that the orbit is a conic of latus rectum , =h 2 lp,, and of arbitrary 

 eccentricity e having the origin as one focus. 



This investigation of the possible central orbits with acceleration /n/r 2 may 

 be taken to replace Newton's investigation, of which a version was given in 

 Article 55. 



2. To find all the orbits which can be described with a central acceleration 

 varying inversely as the cube of the distance. 



We have f=jj.u 3 so that 



- 



There are three cases according as A 2 > = or </*. 



(1) When A 2 >/z, 1 - 2 is positive, put it equal to n 2 . 



Then all the possible orbits are of the form u=A cos(n0 + a), they include 

 a straight line for the case n = 1. 



(2) When h 2 =p we have -v = so that u = A0 + B where A and B are 

 cL6 



arbitrary constants. If A = the orbit is a circle, otherwise it is a hyperbolic 

 spiral, as we see by choosing the constant B so as to write the above 



(3) When W<\L, 1 - 2 is negative, put it equal to n 2 . 



Then all the possible orbits are of the form 



u = Acosh(nd+a) or u = ae n9 + be~ n9 . 

 Putting a or b equal to zero we have an equiangular spiral. 

 3. By integration of the equation 



d 2 u JL 



prove that all the orbits that can be described with a central acceleration 

 proportional to the distance are ellipses having the origin as centre. 



4. If / is any function of r show that one of the possible orbits is a circle 

 described about its centre. 



