LAW OF THE UXIVEBSE. 243 



- , or, because S (the parameter) is 



2 QJ (a -(- X) a \JQ Q" 



constant, inversely as the square of the distance : And the 



other formula F as , 7 gives the same result for the law 

 p 3 dr 



of force, or ,, 3 .* 



4 o 1. 



Again, in the ellipse, if a be half the transverse axis, and b 

 half the conjugate, and r the radius vector, we have p = b 



r & b d r 



, and dp = -; therefore the formula 



2 a - > 



dp ab dr a . v * 



becomes = - , or the force is m- 



p.dr b* *J r X r ? X d r b r* 



versely as the square of the distance. 



Lastly, as the equations are the same for the hyperbola, 

 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 - - becomes , which. 



p 3 E a 4 



being constant, the force is everywhere the same. But if the 

 centre of forces is not that of the circle, but a point in the 



circumference, the force is as . 



r" 



Eespecting centrifugal forces it may be enough to add. 

 that if v is the velocity and r the radius, the centrifugal force 



v* 



/, in a circle, is as . Also if E be the radius of curvature, 

 r 



f for any curve is = ^ . When a body moves in a circle by 

 a centripetal force directed to the centre, the centrifugal force 



* This result coincides with the synthetical solution of Sir Isaac Newton 

 in Prop. XIII. 



R 2 



