11, 12] KEPLER'S LAWS. 19 



and will find the value of the central acceleration. We have 34) 



but from Kepler's first law, 



n dS dy 



2 -7T- = T -3T = const. = h. say, 

 dt dt J ' 



dcp _ h 



Now changing the variable from t to (p, 



dr dr dcp _ h dr _ , d / 1 \ 



dt dcp dt ~ r 2 dcp ~ dcp\r / 



Differentiating by t, 



dt* dcp^ \r / dt r* dq 



From the equation of the path we obtain 

 lie 



\r/~ -7 Sn( P> 

 d 2 /I \ e 1 1 



j ^ I I = COS Op = 



d(p 2 \r/ p p r 



Inserting this value above gives 



d*r _ h* W 



and finally, 



Thus the fact that the path is a conic section shows that the central 

 acceleration varies inversely as the square of the length of the radius 

 vector. The negative sign shows that the acceleration is toward 

 the sun. 



The third law states that for different planets the squares of 

 the times of describing the orbits are proportional to the cubes of 

 the major axes. 



Since 



if T is the time of a complete period JiT is twice the area of the 

 orbit. 



JiT= 



