NEWTOX S PRINCIPIA. 81 



as the square of the distance. This forms the subject of 

 the Eighth Section, which therefore bears to the Third, 

 Fourth, Fifth, and Sixth, the same relation that the con 

 cluding investigation of the Seventh Section (on rectili 

 near motion influenced by centripetal force) bears to the 

 rest of that section. 



The length at which we before went into the solution 

 of the problem of central forces (inverting somewhat the 

 order pursued in the Principia) makes it less necessary to 

 enter fully into the general solution in this place. TTe 

 formerly gave the manner of finding the force from the 

 trajectory in general terms, and showed how, by means 

 of various differential expressions, this process was faci 

 litated. It must, however, be remarked, that the inverse 

 problem of finding the trajectory from the force, is not 

 so satisfactorily solved by m^ns of those expressions. 

 For example, the most general one at which we arrived of 

 vy + ( x - of xdx^.d^i, C 



f + (* ~ *T 



the force inversely as the square of the distance, presents 

 an equation in which it may be pronounced impossible to 

 separate the variables so as to integrate, at least while 



d X, the differential of ~, remains in so unmanageable a 



form; for then the whole equation is , , - -, - x , N , 



2 (ydx (x a) d yj 



= =, and thus from hence no equation to 



(f + (x- of) i 



the curve could be found. It cannot be doubted that Sir 

 Isaac Newton, the discoverer of the calculus, had applied 

 all its resources to these solutions, and as the expressions 



for the central force, whether 3 =~, or 3 ;T &amp;gt; or 



