LAW OP THE UKIVEESE. 251 



of the inverse subduplicate of the distance, or =, and in a 



Vr 



smaller ratio in the latter curve, while in the parabola = 



V r 

 is their exact measure. 



To these useful propositions, Demoivre added a theorem of 

 great beauty and simplicity respecting motion in the 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, a and b the axes, and p the parameter, t = , and 



b* - a p; therefore ab = a */ a p = V 3 X J p; and t = 

 mj a 3 , and t 3 = a 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, d priori, as mathematical 

 truths ; first, the areas proportional to the times if the force is 

 centripetal second, the elliptical orbit, and third, the ses- 

 quiplicate ratio of the times and distances, if the force is 

 inversely as the squares of the distances, 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 sec- 

 tion being to that in a circle at the given distance D as m to 

 n, and the perpendicular to the tangent being p, the lesser 



Imp . 2Dn 2 



axis will be - - , and the greater axis - ; -- ;, tne 



/ o fi ft a V n* - Ml* 



V 2 n* m* z w -m 



