214 NEWTON S PKINCIPIA. 



whence it follows that 



. 

 2 r 



or the density varies inversely as the distance. The nega 

 tive sign shows that the angle a must be greater than a 

 right angle, or that the particle continually approaches the 



centre of force. Let x = , where D is the resistance 



at a unit of distance to a unit of velocity, then 

 ^ ___ cos 

 ~~2~~ 



On looking at equation (2) we see that whatever be the 

 law of force, the velocity is the same as that in a circle at 

 the same distance, and when the force varies inversely as 

 the Square of the distance, we have 



-V?- 



This is Newton s first corollary. 



Again, when K or D is given, equation (3) gives us the 

 means of finding . Thus a spiral may be fitted to any 

 density. This is Newton s second corollary. 



And when the a or the spiral is given, the ratio of the 

 resistance to the centripetal force is easily found. We 

 observe that since cos. a must be less than unity, this ratio 

 must be less than J, otherwise the orbit described will not 

 be the equiangular spiral. When this ratio is \ exactly, 

 the value of a is zero ; that is, the spiral is reduced to its 

 limiting case, viz., a straight line passing through the pole. 

 This includes Newton s third and fourth corollaries. 



Also the time of describing any arc may be found; 

 for since 



