2 3 8.] 



PLANE MOTION. 



I2 9 



From this fact Newton drew the conclusion that in the law 

 of acceleration, 



I J=fV)=% ' . : (62) 



1 the constant /UL has the same value for all the planets. 



Our formulas show this as follows. Let T be the periodic 

 time of any planet, i.e. the time of describing an ellipse whose 

 .semi-axes are a, b. Then, since the sector described in the 

 time T is the area irab of the whole ellipse, we have by (53) 



Substituting in (61) the value of h found from this equation 

 we have 



Hence 



is constant by Kepler's third law. 



(63) 

 (64) 



237. As mentioned in Art. 230, the velocity v can be ex- 

 pressed in terms of the perpendicular/ let fall from the centre 

 on the tangent to the path : 



= 



(65) 



The acceleration / is also conveniently expressible in terms 

 of/. We have by (57) 



dr 



= _i ffi*.(L\ = # 



2 3 



dr 



(66) 



238. Finally, another expression for the acceleration is some- 

 times found convenient. In any motion, the component of the 

 acceleration along the radius vector is (see Art. 161) 



J 



PART I 9 



. <Pr (dOV 



j A*I I 



dt* \dt) 



