256 CENTRAL FORCES; 



iii. The next object of research is to generalise the pre- 

 ceding investigations of trajectories from given forces, and 

 of motion in given trajectories, applying the inquiry to all 

 kinds of centripetal force, and all trajectories, instead of con- 

 fining it, to the conic sections, and to a force inversely as the 

 square of the distance. 



We 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 facilitated. 

 It must, however, be remarked, that the inverse problem of 

 finding the trajectory from the force, is not so satisfactorily 

 solved by means of those expressions. For example, the most 



, ... . , f 



general one at which we arrived ot 



(x a 

 n 



being put = - - -- -, or the force inversely as the square 

 y* + (x a)* 



of the distance, presents an equation in which it may be pro- 

 nounced impossible to separate the variables so as to inte- 



grate, at least while d X, the differential of , remains in 



dx 



so unmanageable a form ; for then the whole equation is 

 d*ydx-d*xdy C 



, 



2(ydx-(x-a)d y y 



, and thus from 



hence no equation to 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 



d T) 



expressions for the central force, whether - , or - , or 



'2p*.r p 3 dr 



d^ x \l s? -4- y 2 

 -- (in some respects the simplest of all, being 



taken in respect of dt constant, and which is integrable in 



