66 ON SOME PROPERTIES OF THE PARABOLA. 



x 

 * m 



we have x v - a (mm - 1), 

 y^ a(m + m), 



_ sin a , _ sin a' 



OOo Ot CyUQ CA 



cos (a + a') 

 x. a , , 



cos a cos a 



sin (a + a') 

 " l cos a cos a' 



To simplify these expressions turn the axes through an 

 angle = (a + a! + a"), and if a?", y" be the new values of the 

 co-ordinates, we find, after some simple reductions, 



a cos a" a sin a" 



y =- 



cos a cos a cos a cos a 



Squaring these and adding, 



"2, " = * 



* ~~ 



cos 2 a cos 2 a! cos a cos a' cos a" cos a cos a 

 aa" 



cos a cos a cos a 



Now this being symmetrical between a, a', a", will hold 

 equally true of the three points of intersection, and it is the 

 equation to a circle passing through the origin which is the 

 focus, whose diameter coincides with the axis of a?, and whose 

 radius is 



2 cos a cos a' cos a" 



The chief advantage of this method besides its simplicity is, that 

 it gives us very readily the radius of the circle, and the position 

 of the diameter which passes through the focus. 



It is easily seen that the distances from the focus of the three 

 points of intersection of the tangents are respectively 

 a r a a 



y n* A. _____________ 



cos a cos a' ' cos a cos a" ' cos a' cos a" " 

 The area of the triangle formed by the intersection of the tan- 

 gents, can be expressed by an elegant symmetrical function of 



