19, 20] CENTRAL FORCES. 39 



It will now be convenient to change the independent variable 

 from t to (p and at the same time to introduce the reciprocal of r, 



^ -, 

 dr dr dtp 



and introducing the value of -^ from 20) gives 



dr li dr , du 



dt r z dcp dcp 



Differentiating again and proceeding in like manner, 



so that finally, 



<T-I\ 



21) 



d*r ^ d*u dcp 



= h-= -f = 

 dt* dcp z dt 



If a r is given as a function of the distance, this is the differential 

 equation of the path. As an example let us consider attractions 

 varying according to the Newtonian law. We have then 



and the differential equation becomes 



*> . . S+= 



or as we may write it, 



Thus w.~ ! TT is given in terms of qp by an equation like equation 8), 

 whose integral is 



u i = a cos (cp a) 

 fi 



... ah* 

 or putting = e, 



This is the equation of a conic section with which we started 

 the investigation of 12. In order to find the eccentricity e let us 

 consider the initial circumstances, or the magnitude and direction of 

 the velocity for a given position of the body. Let the body be 

 projected from a point <p = 0, r = R with a velocity F, making an 

 angle s with the radius vector. Now we have 



., dcp 1 dr 1 du 



24) tan s = r -~j cot s = -v- = ^ 



dr r dcp u dcp 



