LAW OF INVERSE SQUARE 277 



and equation (111) becomes 



, , .7/3 hdr 



whence we obtaui du - 



*L- 

 r h 

 giving, on integration, 6 = sin" 1 + e, 



where e is a constant of integration. 

 Simplifying, this becomes 



'-sin(0-e), 

 and if we compare with the equation 



1 = e cos 0, 



T 



we see that equation (113) represents a conic, having the origin 

 as focus, and being of semi-latus rectum I and eccentricity 

 e = -xjl H In order that the line = may coincide with 



TT 



the major axis of the conic, the value of e must be 



2 



223. We notice that if 



E is positive, then e > 1, and the orbit is a hyperbola ; 

 E is zero, then e = 1, and the orbit is a parabola ; 

 E is negative, then e < 1, and the orbit is an ellipse. 



Thus the class of conic described depends solely on the value 

 of E, and not. on that of h. And it should be noticed that, if we 

 are given the point of projection of a particle, and also its velocity 

 of projection, the value of E is determined, for by equation (110) 



