60 



the subduplicate ratio of the parameter inversely, is in the 

 sesquiplicate ratio of the transverse axis, and equal to the 

 periodic time in a circle whose diameter is that axis. It 

 is also easy to show from the formula already given re 

 specting the perpendicular to the tangent, that the velocities 

 of bodies moving in similar conic sections round the same 

 focus, are in the compound ratio of the perpendiculars in 

 versely and the square roots of the parameters* directly. 

 Hence in the parabola a very simple expression obtains for 

 the velocity. For the square of the perpendicular being 

 as the distance from the focus by the nature of the curve 

 (the former being a 2 + a x, and the latter a + x), the 

 velocity is inversely as the square root of that distance. In 

 the ellipse and hyperbola where the square of the per 

 pendicular varies differently in proportion to the distance, 

 the law of the velocity varies differently also. The square 

 of the perpendicular in the ellipse (A being the transverse 

 axis and B the conjugate, and r the radius vector) is 



B 2 x r B 2 x r 



-r- ; in the hyperbola, -r , or those squares of 



xx T J\. -J- 7* 



the perpendicular vary as and . in those 



A r A + r 



curves respectively, B 2 being constant. Hence the ve 

 locities of bodies moving in the former curve vary in a 

 greater ratio than that of the inverse subduplicate of the 



distance, or _, and in a smaller ratio in the latter curve, 

 Vr 



while in the parabola -=. is their exact measure. 



v r 



To these useful propositions, Demoivre added a theorem 

 of great beauty and simplicity respecting motion in the 



* By parameter is always to be understood, unless otherwise mentioned, 

 the principal parameter, or the parameter to the principal diameter. 



