1887.] A. Mukhopadhyay — Memoir on Flmie Analytic Geometry, 313 



we see that it is the equation to a conic which is an ellipse, a parabola, 

 or an hyperbola according as 



A2 + B2 Z = T 1. 

 Now, knowing that the curve is a conic, we may next compare its equa- 

 tion with the focal polar equation 



Z=:p (1 + e cos 6). 

 Remembering that p is a function of x and y, we conclude that the 

 absolute terms in the two equations must be identical, whence 



C = l = semi-latus-rectum. 

 Again, as the equation may be written in the form 



where p is the distance of any point on the curve from a fixed point, and 



Ax + By + G 



v/am-'b3 



is the perpendicular on the line Ax + By + = 0, we see, by attending to 

 the focus-directrix method of generating conies, that the curve is a conic 

 of which the directrix is 



Ax+By + C = 0, 

 and the eccentricity is given by 



e8 = A2 + B«. 

 §. 18. Elliptic Motion. — In order to represent these properties 

 geometrically, and to shew their relation to elliptic motion, it is con- 

 venient to begin with the following method of integrating the equations 

 of motion. We have, as usual, 



dPx fix 



d^y _ _Fy 





dt^" rS' 





dy dx , 







Now 



- == COS 6, - = sin ^; 

 r r 



therefore 



d /x\ . , de y 

 Tt\-r) = -''''^'Tt='V^-^^ 



whence 



y \ d /x\ 

 7^^~hlt\7r 



and, similarly. 



X I d /y\ 



r^~ hdt\r)' 



40 





