50 NEWTON S PRINCIPIA. 



Lastly, as the equations are the same for the hyper 

 bola, with only the difference of the signs, the value 

 of the force is also inversely as r 2 , or the square of the 

 distance. In the circle a = the radius = r = p ; hence 



M ft 



becomes -^-, which, being constant, the force is every- 



where the same. But if the centre of forces is not that 

 of the circle, but a point in the circumference, the force is 



1 

 V 



Respecting centrifugal forces it may be enough to 

 add, that if v is the velocity and r the radius, the 



v 2 

 centrifugal force f, in a circle, is as . Also if R be 



v 1 

 the radius of curvature, f for any curve is = ^ 



When a body moves in a circle by a centripetal force 

 directed to the centre, the centrifugal force is equal 

 and opposite to the centripetal. Also the velocity in 



uniform motion, like that in a circle, being as --, the 



6 



space divided by the time, and the arc being as the 



s 2 r 



radius r, f is as - ^ or as ~^. If two bodies moving in 



/* 6 C 



different circles have the same centrifugal force, then the 

 times are as ^ r. It is to the justly celebrated Huygens 

 that we owe the first investigation of centrifugal forces. 

 The above propositions, except the second, are abridged 

 from his treatise.* 



- The rest of the investigation of centripetal forces is an 

 expansion of the formulas above given, and their appli 

 cation to various cases, but chiefly to the conic sections. 

 It may be divided into four branches. First, the rules 



* Horologium Oscillatorium, ed. 1673, p. 159, App. 



