LAW OF THE UNIVERSE. 253 



certainty as that of untroubled motion round a point by virtue 

 offerees directed thither.* 



We have thus seen how important in determining all the 

 questions, both direct and inverse, relating to the centripetal 

 force, are the perpendicular to the tangent and the radius of 

 curvature. Indeed it must evidently be so, when we con- 

 sider, first, that the curvature of any orbit depends upon the 

 action of the central force, and that the circle coinciding with 

 the curve at each point, beside being of well-known proper- 

 ties, is the curve in which at all its points the central force 

 must be the same ; and, secondly, that the perpendicular to the 

 tangent forms one side of a triangle similar to the triangle 

 of which the differential of the radius vector is a side ; the 

 other side of the former triangle being the radius vector, 

 the proportion of which to the force itself is the material 

 point in all such inquiries. The difficulty of solving all these 

 problems arises from the difficulty of obtaining simple ex- 

 pressions for those two lines, the perpendicular p and the 

 radius of curvature R. The radius vector r being always 

 V x* -f if interposes little embarrassment ; but the other two 

 lines can seldom be concisely and simply expressed. In 

 some cases the value of F, the force, by dr and dp may be 

 more convenient than in others ; because p may involve the 

 investigation in less difficulty than E ; besides that p 8 enters 

 into the expression which has no differentials. But in the 

 greater number of instances, especially where the curve is 



r 

 given, the formula -^-j- will be found most easily dealt with. 



J) -tw 



ii. We are next to consider the motion of bodies in conic 

 sections which are given, and ascending or descending in 

 straight lines under the influence of gravity ; that is, the 

 velocities and the times of their reaching given points, or 



* Laplace (Mec. Cel. lib. xv. ch. i.) refers to this remarkable passage as 

 the germ of Lagrange's investigations in the Berlin Memoires for 1786. 



