NEWTON S PRINCIPIA. 215 



dr 



- = v COS a 

 a t 



- COS a 



t = CSrdr 

 COS a &quot; &amp;lt;\//*t/ 



So that the time of going from the distance r l to the dis 

 tance i\ is 



E-fJ^jjL (r^-r^) - (4). 



When a is nearly a right angle, this time becomes very long. 

 If a = 0, the same formula will give the time of descent 

 T down any part of a radius vector. Hence we see 

 that 



ay 



T = , 



COS a 



and the number of revolutions described may also be 

 found : for if be the angle the radius vector r makes 

 with any fixed straight line, we have 



it dr . 



d V = tan a 



r 



^2 ~~ ^1 = tan a 10 &quot;&quot;* 



T \ 



Hence the number of revolutions will be 

 ,, T tan a , r 



/e ,\ 

 (o). 



This includes Newton s fifth and sixth corollaries. 



If we neglect the eccentricities of the planetary orbits, 

 the velocity at any point in vacuo is given by the usual 

 formula 



u 2 = Pr 



Let us then assume as the velocity in a resisting medium 

 v 2 = Pr ( 



P 4 



