KINEMATICS. [239. 



As in our problem the total acceleration is along the radius 

 vector, in the sense towards the origin, we have 



or since, by (51), 



The first term is what the acceleration would be if the motion 

 were rectilinear along the radius vector ; the second term 

 represents what is due to the curvature of the path. 



239. Planetary Motion, in its simplest form, is (see Art. 223) 

 that particular case of central motion in which the acceleration 

 is inversely proportional to the square of the distance from the 

 centre O, so that 



where //, is a constant, viz. the acceleration at the distance r= I 

 from O. 



The equations of motion (49) are in this case, with O as 

 origin, 



d^x _ _ x_ d^y _ _ y_ /^gx 



Combining these by the principle of energy (Arts. 231, 232), 

 we find 



dt r s \ dt ' dt) r* dt\ 2 



< 



_ 



r*dt dt ' 



hence integrating 



-IL (69) 



T rn 



