86 NEWTON S PRINCIPIA. 



a - D 



x arc cos. , _ 



+ 2 Q 2 V2 c - x 



And if we get D from this, in terms of cos. x 9 we have 

 then to obtain P C by similar triangles, and from I P C 

 being right-angled and I C = x, to obtain P I, in order to 

 have the curve V I K. 



But if we proceed otherwise, and instead of working 

 by quadratures, take v the velocity of the body at I, or 



gi 



in the straight line at D, and make ^ the area described 

 in a second, and the angle V C I, we obtain as a 



(* d 3T 



polar equation to V I K, d = =. (x being 



x v 4 x 1 v 2 c 2 



in this case both C D and the radius vector). Then, 

 to apply this general equation to the case of the centri 



petal force being as , let the force at the distance 1 be 



X 



put equal to unity, and supposing the velocity of pro 

 jection to be that acquired in falling from an infinite 

 height, the equation to the trajectory becomes 

 c d x 



, . . , 



and interatin, = 



2 



, . 



x Vl c 2 v 4 c 



, x 

 x log. -. 



The whole subject of centripetal forces, inverse and 

 direct, under the four heads which we began by stating, 

 has therefore been (Jiscussed, but always upon the assump 

 tion that the bodies acted upon move in orbits which 

 remain at rest, and thus that the axis of the curve which 

 they describe remains constantly in the same position. 

 Another subject of inquiry is presented to us if that axis 



