I 3 i.] CENTRAL FORCES. 69 



with one of the foci at the centre of force, the conic being an 

 ellipse, parabola, or hyperbola, according as 



V| 2 f (13) 



> r o 



If the force be repulsive, the same reasoning will apply, except 

 that IJL is then a negative quantity. The orbit is, therefore, in 

 this case always hyperbolic ; the branch of the hyperbola that 

 forms the orbit must evidently turn its convex side towards the 

 focus at which the centre of force is situated, since the force 

 always lies on the concave side of the path. 



131. To exhibit fully the determination of the constants and the 

 dependence of the nature of the orbit on the initial conditions, a solution 

 somewhat different from that given in kinematics will here be given for 

 the problem of planetary motion in its simplest form. 



With/(r)=/x/V 2 , the equation of kinetic energy, (7), Art. 112, gives 



or, if the constant of integration be denoted briefly by h and u=i/rbe 

 introduced : ' 



v* = 2fjiU + h, where h = v - (14) 



r o 



Substituting this expression of i? into the equation (9), Art. 113, we 

 find the differential equation of the orbit in the form 



' 2 



or 



To integrate, we introduce a new variable ' by putting 



the resulting equation, 



