252 CENTRAL FORCES; 



signs being reversed in the denominator of each quantity for 

 the case of the hyperbola. Hence the very important con- 

 clusion that the length of the greater axis does not depend at 

 all upon the direction of the tangential or projectile force, 

 but only upon its quantity, the direction influencing the 

 length of the lesser axis alone. 



Lastly, it may be observed, that as these latter propositions 

 give a measure of the velocity in terms of the radius vector 

 and perpendicular to the tangent for each of the conic sec- 

 tions, we are enabled by knowing that velocity in any given 

 case where the centripetal force is inversely as the square of 

 the distance, and the absolute amount of that force is given, 

 as well as the direction of the projectile force and the point 

 of the projection, to determine the parameters and foci of the 

 curve, and also which of the conic sections is the one de- 

 scribed with that force. For it will be a parabola, an hy- 

 perbola, or an ellipse, according as the expression obtained 

 for p* (the square of the perpendicular to the tangent) is as 

 the radius vector, or in a greater proportion, or in a less pro- 

 portion. This is the problem above referred to, which John 

 Bernoulli had entirely overlooked, when he charged Sir 

 Isaac Newton with having left unproved the important 

 theorem respecting motion in a conic section, which is clearly 

 involved in its solution. 



Before leaving this proposition, it is right to observe that 

 the two last of its corollaries give one of those sagacious 

 anticipations of future discovery which it is in vain to look 

 for anywhere but in the writings of Newton. He says, that 

 by pursuing the methods indicated in the investigation, we 

 may determine the variations impressed upon curvilinear 

 motion by the action of disturbing, or, what he terms, foreign 

 forces ; for the changes introduced 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 Laplace, whose immortal labours have 

 reduced the theory of disturbed motion to almost as great 



