18 I KINEMATICS OF A POINT. LAWS OF MOTION. 



If the motion is not in one plane, we have, differentiating the sector 

 velocity components 24) 



The resultant of these is the moment of the acceleration. The fact 

 that the moment of the acceleration is the exact time derivative of 

 the moment of velocity leads to an important general principle of 

 mechanics, the so-called Law of Areas. 



12. Kepler's Laws. We may now obtain Newton's conclusions 

 from Kepler's three laws of planetary motion, which were purely 

 kinematical and hased on a great amount of observational material 

 collected by Tycho Brahe. The first law states that the areas swept 

 over by the radius vector drawn from the sun to a planet in equal 

 times are equal. (The motion is in one plane.) That is 



dS 



~dt = C0nst > 



dt* 



therefore from 37) the moment of the acceleration with respect to 

 the sun is zero. Consequently the line of direction of the accelera- 

 tion passes through the sun, or the acceleration is central. 



The second law states that the planets describe ellipses about 

 the sun as a focus. The ellipse being always concave toward the 

 focus, the acceleration is directed toward tbe sun. In order to 

 deduce the quantitative meaning of the second law, we will use the 

 polar equation of a conic section referred to the focus, 



1 -j- e cos cp 



1) If d is the distance from focus to directrix, e the eccentricity, by the 

 definition of a conic section, 



r ed = p 



= e, r = * 

 d r cos cp 1 -f- e cos cp 



When cos cp = 1 



~l + e 



COS qp = 1, 



