1887.] A. Mukhopadliyay — Memoir on Plane Analytic Geometry. 345 



Imagine a particle to describe the parabola under tlie action of a force 

 directed to the new origin as centre ; and suppose it to be started from 

 the apse with the velocity in a circle at the same distance. Then 



^ dy ^ dx 



•^ dt dt' 



/dyV dhj ^ d^x 



dii 

 whence {xAr^na) -7- = — ^. 



av 



Therefore — -— _ P ^ = - 2a. P. -, 



\x i- Zna) •* r r 



where P is the central force. 

 This may be written 



/^^a p p 



{x-\-1nay r ^'^ ^ r ^ ^ 



which gives 



P=^ L_^ (182) 



2a {x-Y^nay 

 But aj" + ^/ = r^ 



y^ =:4ia (x-\-na). 

 Eliminating y, this gives a quadratic for x, whence we derive 



JL 



x-J[-2na = 2a (n-l)+ | r^H-4a2 (1 -n)| \ 

 Substituting in (182), we get 



- ^' ' (183) 



2a 



2a (n - 1) + ^7'-' + 4a2 (1 - n) 



which gives the law of force in terms of the radius vector. For an 

 interesting discussion of a kinetic difficulty in connection with this 

 dynamical problem, see a note by Dr. Besant in the Quarterly Journal of 

 Mathematics, t. XI, 38. 



§.31. Geometrical Applications.— Thus far we have solved a 

 purely dynamical question ; we now proceed to obtain some interesting 

 geometrical properties of the parabola. We have 



]j^ dr 2 dr \2^^/' 



44 



