NEWTON S PRINCIPIA. 63 



duced by these in some places, he says, may be found, and 

 those in the intermediate places supplied, by the analogy 

 of the series. This was reserved for Lagrange and La 

 place, whose immortal labours have reduced the theory of 

 disturbed motion to almost as great certainty as that of 

 untroubled motion round a point by virtue of forces di 

 rected thither.* 



We have thus seen how important in determining all the 

 questions, both direct and inverse, relating to the centri 

 petal force, are the perpendicular to the tangent and the 

 radius of curvature. Indeed it must evidently be so, when 

 we consider, Jirst, 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 properties, is the curve in which at all its points 

 the central force must be the same; and, secondly r , 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 it 

 self is the material point in all such inquiries. The difficulty 

 of solving all these problems arises from the difficulty of 

 obtaining simple expressions for those two lines, the per 

 pendicular p and the radius of curvature K. The radius 

 vector r being always */x~+y 2 interposes little em 

 barrassment; but the other two lines can seldom be con 

 cisely and simply expressed. In some cases the value of 

 F, the force, by d r and dp may be more convenient than 

 in others; because p may involve the investigation in less 

 difficulty than K; besides that p 3 enters into the expression 

 which has no differentials. But in the greater number of 



* 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. 



