40 H. PARTICULAR MOTIONS OF A POINT. 



Putting <p = in equation 23) gives 



-jg- = jt (1 + e cos a) or 



25) -^ -- l=ecosa. 



Differentiating equation 23) 



du y / \ 



Introducing this into 24) and putting cp = 0, 



JJye . 

 cot a = -- jry- sin a, or 



0/ x . ft 2 COt 8 



Squaring and adding 25) and 26) 

 27) e 2 = ^cosec^- 



Also dividing 26) by 25) 



OON ft 2 cots 



28) 



Ey-h* 



Now h being the constant moment of velocity ( 8), is equal to the 

 value when cp = 0, 



29) 

 Inserting this in 27) and 28) gives 



30) 

 31 ) 



2 v 



According as F 2 is less than, equal to, or greater than --> e will 



be less than, equal to, or greater than 1, and the orbit will be 

 respectively an ellipse, parabola, or hyperbola. 



The critical velocity, F, has a simple physical significance. 

 Suppose we consider a particle falling from infinity straight toward 

 the center of attraction. Its equation of motion is 



d*r _ j_ 



dt* ~ ~ T 2 " 



Multiply by -j-> both sides become exact derivatives and we may 

 integrate, obtaining 



