:ii6.] CENTRAL FORCES. 6l 



When the equation of the orbit is known, i.e. when r is known 

 .as a function of 9 or vice versa, the time t of the motion can be 

 found by integrating equation (4), viz. 



dt=-r*>dQ. 

 c 



115. If the second expression for v 2 in (9) be introduced into 

 the differential equation of kinetic energy (6), we find 



or 



This will generally be found the most convenient form for find- 

 ing the law of force when the polar equation of the orbit is 

 given. Again, when/(r) is given, the integration of this differ- 

 ential equation of the second order is often more convenient 

 for finding the equation of the orbit than the method indicated 

 in Art. 114. 



It may be noted that the important relation (10) can be de- 

 rived directly from the equations of motion (3), by eliminating / 

 by means of (4) and introducing u for i/r. We have 



^_ _ 2 2 

 dt~ ' d&^ 



If these values be substituted into the first of the equations (3), 

 the relation (10) will result. 



116. When the equation of the orbit can be expressed con- 

 veniently in terms of r and/, as is, for instance, the case for the 

 conic sections, it is of advantage to combine the equation of 

 kinetic energy, </(J^ 2 ) = f(r)dr, directly with the equation 

 resulting from the principle of areas, pv c. This gives 



dr ~2 dr ' f dr 



