12 8 KINEMATICS. [234. 



where u\/r. Substituting for v its expression in terms of 

 r or #, from (58), we have the differential equation of the path 

 which is directly integrable. 



Shorter methods will often suggest themselves in particular 

 cases. 



234. To solve the -converse problem, viz. to find the law of 

 acceleration when the equation of the path is given, we havei 

 only to substitute in (57) the expression of ^ from (59). We 

 find, with u\lr t 



dr du dr du 



235. Kepler in his second law had established the empirical 

 fact that the orbits of the planets are ellipses, with the sun at 

 one of the foci. 



From this, Newton concluded that the law of acceleration 

 must be that of the inverse square of the distance from the sun. 



Equation (60) allows us to draw this conclusion. The polar 

 equation of an ellipse referred to focus and major axis is 



where t=d >2 /a=a(ie 2 ); a, b being the semi-axes, /the semi- 

 latus rectum, and e the eccentricity of the ellipse. Hence 



.and (60) becomes 



236. The third law of Kepler, found by him likewise as an 

 empirical fact, asserts that the squares of the periodic times of 

 -different planets are as the cubes of the major axes of their orbits. 



