60 KINETICS OF A PARTICLE. [113, 



whence i^v^ 2j f(r}dr\ (7). 



i.e. the velocity at any distance r depends only on this distance 

 (besides the initial radius vector and velocity), and is indepen- 

 dent of the path described, being the same as if the particle 

 had been projected with the initial velocity along the straight 

 line joining the initial position to the centre. 



113. To perform the second integration we have only to 

 substitute in (7) for v its value in terms of r and / or r and 6. 

 Now the general expression for the velocity in any curvilinear 

 motion is (Part I., Art. 142) 



dt \dt \dt ddJ 



From these expressions one -of the variables and t can be 

 eliminated by substituting for d6/dt its value c/r* from (4) ; this 

 gives 



It is often convenient to replace the radius vector r by its- 

 reciprocal u=i/r] we then have 



l~ / J. \ 9 ~l 



(9) 



114. The formulas (4) and (7), together with the expressions 

 (8) or (9), contain the complete solution of the two principal 

 problems mentioned in Art. no. Thus, if the law of force be 

 given, the form of the function f(r) is known, and v can be 

 found from (7) in function of r or u ; substituting this value of 

 z; in either (8) or (9), we have a differential equation of the first 

 order between r and /, or between r and 0. The integration of 

 the latter equation gives the integral equation of the orbit. 



On the other hand, if the equation of the path be given, the 

 expressions (8) or (9) furnish the value of v 2 , which, substituted 

 in (6), determines the law of force f(f). 



