314 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



The equations of motion, therefore, become 



fi d (y 



dtZ 



h. dt 



Integrating, we get 



dhj _ (x d /x\ 

 m"^ hdi\r/' 



^ A^ (y __ \ 



dt h \r 7' 



and since 

 we have 



which leads to 



dy 



dt 



dy dx 



^^r-Vn-y)-'' 



which is Laplace's equation. Comparing this with the form 



we find, as it ought to be, 



C = — = semi'latus-rectum. 



This shews why, in integrating the equation 



dr = Xdx + ydy^ 



Laplace at once puts — for the constant of integration. 



§. 19. Geometric interpretation, — The subject may be made 

 still clearer by the 

 help of a diagram. 

 The ellipse is ori- 

 ginally referred to 

 rectangular axes 

 through the focus 

 S ; suppose that the 

 coordinate axes re- 

 volve round the 

 origin, making an 

 angle XS^ ( = 0) 

 with the former po- 

 sition. Then, we 

 have 



