34 On the Property of the Parabola 



In order that the circle may pass through the tliird poin 

 we must have 



or —^ + (yi + ya) c^^sfl + « + wij = 0, which determines a. 

 4 W'tj 



Hence the equation to the circle is 

 x« + 2 ^3/ cosfl + / - (^-^ + (yi + y<,) cos5 + m,) 



we get 



When it cuts the parameter x — m^. Substituting which 

 t 



y^+ (27»'cosfi-^-^jj/-7Wj(y,4-y2)cosa = 0; 



the roots of which are obviously — 2 m, cos 5, and ^^—~^. 



The first of these shows that it passes through the focus. 



We thus see that by the use of oblique coordinates the pro- 

 blem is simplified in the most remarkable manner, and there 

 is no necessity for employing the troublesome method of find- 

 ing the focus, which Mr. Lubbock's solution requires. 



As in some degree connected with this subject, I would call 

 your attention to two properties of the conic sections, which 

 seem scarcely, if at all, known, and are yet deserving of atten- 

 tion. The first is due to Keil, and furnishes an easy method 

 of finding the centre of curvature of a conic section. 



Fig. 2. 



Let A F N be the axis ; F the focus ; P N a normal meet- 

 ing the axis in N ; N Q a perpendicular to the normal meet- 



