NEWTON S PRINCIPIA. 61 



ellipse. The velocity in any point P is to the velocity in 

 T, the point where the conjugate axis cuts the curve, as 

 the square root of the line joining the former point P 

 and the more distant focus, is to the square root of 

 the line joining P and the nearer focus. It follows from 

 these propositions that in the ellipse, the conjugate axis 

 being a mean proportional between the transverse and 

 the parameter, and the periodic time being as the area, 

 that is as the rectangle of the axes directly, and the 

 square root of the parameter inversely, t being that time, 



ab 

 a and b the axes, and p the parameter, t = =, and 



b 2 = ap\ therefore ab = a^ap A/a 3 * ^ p\ and t = 

 Va?, and / 2 = 3 ; or the squares of the periodic times 

 are as the cubes of the mean distances. So that all 

 Kepler s three laws have now been demonstrated, a priori, 

 as mathematical truths ; first, the areas proportional to the 

 times if the force is centripetal second, the elliptical orbit, 

 and third, the sesquiplicate ratio of the times and dis 

 tances, if the force is inversely as the squares of the dis 

 tances, or in other words if the force is gravity. 



Again, if we have the velocity in a given point, the 

 law of the centripetal force, the absolute quantity of 

 that force in the point, and the direction of the projectile 

 or centrifugal force, we can find the orbit. The velocity 

 in the conic section being to that in a circle at the given 

 distance D as m to n, and the perpendicular to the tangent 



beino 1 p, the lesser axis will be -. and the 



O * ^ /r\OO 



v 2n 2 m 2 

 greater axis 2 %&amp;gt; * ne s ig ns being reversed in the 



denominator of each quantity for the case of the hyperbola. 

 Hence the very important conclusion that the length of 



