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



and the complete primitive is (by Article 47) 



v=Acos(d 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 ?, = A 2 //i, and of arbitrary 

 eccentricity e having the origin as one focus. 



This investigation of the possible central orbits with acceleration /i/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 =t 3 so that 



+ 



There are three cases according as A 2 &amp;gt; = or &amp;lt;/x. 



(1) When A 2 &amp;gt;/i, 1 -^ 2 is positive, put it equal to n 2 . 



Then all the possible orbits are of the form u = A cos(w0 + a), they include 

 a straight line for the case n = l. 



(2) When k 2 =u we have -,-2 = so that u = A6 + B where A and B are 



d6 2 



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 



u=A(6-a). 



(3) When A 2 &amp;lt;/&, 1 - jr 2 is ne g ative &amp;gt; P ut ^ e( l ual to ~ n - 



Then all the possible orbits are of the form 



u = Acosh(n6 + a) or u = ae ne + be~ ne . 

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

 3. By integration of the equation 



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. 



