NEWTON S PRINCIPIA. 213 



angle with the radius vector : let this angle be called a. 

 Let r and v be the radius vector and velocity of the particle 

 at any time t. Let P be the central force, and x v- the 

 resistance at any time, then P and x are functions of r. 

 The equation giving the motion along the arc is clearly 



d v _&amp;gt; 



= x v 2 P cos a. 

 a t 



But in all curves 



dr 



- = v COS a 

 a t 



d v d v 



. . -j- = -j- . V COS a 



d e? r 



Hence the above equation becomes 



d r cos a 



It is to be observed that this equation is true whatever 

 be the nature of the curve described. 



The equation giving the motion perpendicular to the arc 

 is well known to be 



^ = P Sin * 

 H 



But in the equiangular spiral the radius of curvature R 



is 7^ , hence we have 



Sin a 



v* = Pr - - (2.) 



If we substitute this in equation (1) we get 

 1 



^dP = 



P \ cos *J r 



f*f O v r&quot; 



.-. log P = C - _ 



c/ V cos a/ r 



which is the required connection. 



If the central force vary inversely as the Square of the 

 distance, we have 



P 3 



