NEWTON S PRINCIPIA. 59 



solution of the inverse problem given by Herrman in the 

 same volume of the Memoires, and which had been com 

 municated to him before it was presented to the Academy. 

 This solution proceeds upon his general expression for the 



j2 x 



centripetal force, - - -v^-j-y 2 ; and the objection made 



is that he works the problem (as he does in a few lines) 

 by multiplications and divisions which show that he was 

 previously aware of the solution in the case of the conic 

 sections. But this is no objection to a solution which being 

 of a problem already known, can only be regarded as a 

 demonstration that the former solution was exact. It is 

 an objection which, if valid, applies certainly to the de 

 monstration which we have just given of the proposition ; 

 but so it does to all the demonstrations of the ancient 

 geometrical analysis. It is a more substantial objection 

 that Herrman omitted a constant in his integration ; but 

 by adding it, Bernoulli shows that the equation which 

 Herrman found, when thus corrected, expresses the conic 

 sections generally. 



This truth, therefore, of the necessary connexion be 

 tween motion in a conic section and a centripetal force 

 inversely as the square of the distance from the focus, is 

 fully established by rigorous demonstration of various kinds. 



If we now compare the motion of different bodies in 

 concentric orbits of the same conic sections, we shall find 

 that the areas which, in a given time, their radii vectores 

 describe round the same focus, are to one another in the 

 subduplicate ratio of the parameters of those curves. From 

 this it follows, that in the ellipse whose conjugate axis 

 is a mean proportional between its transverse axis and 

 parameter, the whole time taken to revolve (or the periodic 

 time) being in the proportion of the area (that is in the 

 proportion of the rectangle of the axes) directly, and in 



